Transport properties across the many-body localization transition in quasiperiodic and random systems

We theoretically study transport properties in one-dimensional interacting quasiperiodic systems at infinite temperature. We compare and contrast the dynamical transport properties across the many-body localization (MBL) transition in quasiperiodic and random models. Using exact diagonalization we compute the optical conductivity $\sigma \left(\omega \right)$ and the return probability $R\left(\tau \right)$ and study their average low-frequency and long-time power-law behavior, respectively. We show that the low-energy transport dynamics is markedly distinct in both the thermal and MBL phases in quasiperiodic and random models and find that the diffusive and MBL regimes of the quasiperiodic model are more robust than those in the random system. Using the distribution of the dc conductivity, we quantify the contribution of sample-to-sample and state-to-state fluctuations of $\sigma \left(\omega \right)$ across the MBL transition. We find that the activated dynamical scaling ansatz works poorly in the quasiperiodic model but holds in the random model with an estimated activation exponent $\psi \approx 0.9$. We argue that near the MBL transition in quasiperiodic systems, critical eigenstates give rise to a subdiffusive crossover regime on finite-size systems.

Figure 1

Left panel: disorder-averaged adjacent gap ratio $\overline{r}$ vs disorder strength $\lambda$ for different system size $L$ for (a) AA model with an open boundary condition (obc), (c) AA model with a periodic boundary condition (pbc), and (e) random model with a pbc. The blue and red dashed lines denote the values of $\overline{r}$ for Gaussian orthogonal ensemble (GOE) and Poisson distributions, respectively. Right panel: disorder-averaged half-chain entanglement entropy $\overline{S}/S_T$ vs disorder strength $\lambda$ for different system size $L$ for (b) AA model with a obc, (d) AA model with a pbc, and (f) random model with a pbc. All quantities are calculated from the middle $\frac{1}{3}$ of the spectrum. Inset: zoom-in plot near MBL transition.

Figure 2

Disorder-averaged dc conductivity ${\sigma }_{\mathrm{dc}}$ vs system size $L$ for different disorder strength $\lambda$ for (a) AA and (b) random models. The dc conductivity is calculated from the middle $\frac{1}{3}$ of the spectrum.

Figure 3

Disorder-averaged optical conductivity $\sigma \left(\omega \right)$ vs frequency $\omega$ of (a) AA and (b) random models with system size $L=16$ for different disorder strength $\lambda$. The dashed line is the fit to the equation $\sigma \left(\omega \right)\sim {\omega }^{\alpha }$. Disorder-averaged return probability $R\left(\tau \right)$ vs time $\tau$ of (c) AA and (d) random models with system size $L=16$ for different disorder strength $\lambda$. The dashed line is the fit to the equation $R\sim {\tau }^{-\beta }$. All the quantities are computed from the middle $\frac{1}{3}$ of the spectrum.

Figure 4

Top panel: the exponent (a) $\alpha$ of $\sigma \left(\omega \right)\sim {\omega }^{\alpha }$ and (b) $\beta$ of the return probability $R\sim {\tau }^{-\beta }$ for the AA model with different system sizes. Middle panel: the exponent (c) $\alpha$ and (d) $\beta$ for the random model with different system sizes. Bottom panel: comparison between (e) $\alpha$ and (f) $\beta$ for AA and random models for $L=16$.

Figure 5

Probability density functions of dc conductivity $P\left({\sigma }_{\mathrm{dc}}\right)$ of AA (top panel) and random (bottom panel) models for $L=12$ and different disorder strength $\lambda$. The vertical solid lines denote the median of the distributions.

Figure 6

Comparison between the probability density functions of dc conductivity $P\left({\sigma }_{\mathrm{dc}}\right)$ of AA (solid line) and random (dashed line) models for $L=12$ and different disorder strength $\lambda$. The vertical solid and dashed lines denote the corresponding median value of the distributions. The dc conductivity is calculated from the middle $\frac{1}{3}$ of the spectrum.

Figure 7

Probability density functions of dc conductivity $P\left({\sigma }_{\mathrm{dc}}\right)$ of AA (left panel) and random (right panel) models for different system size $L$ and disorder strength $\lambda$ in diffusive (top panel), near the transition (middle panel), and MBL (bottom panel) regions.

Figure 8

Full width at half-maximum over samples of the probability distribution of ${\sigma }_{\mathrm{dc}}$ $\left[{\mathrm{FWHM}}_{\mathrm{sample}}\left({\sigma }_{\mathrm{dc}}\right)\right]$ vs disorder strength $\lambda$ for (a) AA and (b) random models. Standard deviation of ${\sigma }_{\mathrm{dc}}$ over eigenstates $\left[{\mathrm{\Delta }}_{\mathrm{state}}\left({\sigma }_{\mathrm{dc}}\right)\right]$ vs disorder strength $\lambda$ for (c) AA and (d) random models.

Figure 9

Plots of $\mathrm{\Delta }\alpha$ and ${\mathrm{\Delta }}_S^{\mathrm{samples}}$ vs $\lambda$ for (a) AA and (b) random models. $\mathrm{\Delta }\alpha ={\alpha }_{\mathrm{small}}-{\alpha }_{\mathrm{large}}$ is the difference between the power-law exponent $\alpha$ of sample averaged $\sigma \left(\omega \right)$ having ${\sigma }_{\mathrm{dc}}<{\sigma }_{\mathrm{dc}}^{\mathrm{median}}$ $\left({\alpha }_{\mathrm{small}}\right)$ and ${\sigma }_{\mathrm{dc}}>{\sigma }_{\mathrm{dc}}^{\mathrm{median}}$ $\left({\alpha }_{\mathrm{large}}\right)$ with ${\sigma }_{\mathrm{dc}}^{\mathrm{median}}$ being the median value of the dc conductivity. We compare this $\mathrm{\Delta }\alpha$ with the standard deviation of half-chain entanglement entropy over different samples $\left({\mathrm{\Delta }}_S^{\mathrm{samples}}\right)$ and find that the region of $\lambda$ over which the entanglement entropy and transport are dominated by the sample-to-sample variations are similar. The value of $\mathrm{\Delta }\alpha$ is normalized by 2, which is the difference between $\alpha$ for the pure thermal and pure MBL region, while the value of ${\mathrm{\Delta }}_S^{\mathrm{samples}}$ is normalized by $S_T=0.5[L\ln \left(2\right)-1]$, which is the entanglement entropy of a randomly drawn pure state.

Figure 10

Activated dynamical scaling of $\log \left[\sigma \right(\omega )/\omega ]$ vs $\text{log}\left(\omega \right)(|\lambda -{\lambda }_c{|/{\lambda }_c)}^x$ for (a) AA and (b) random models with $L=16$. The conductivity $\sigma \left(\omega \right)$ is calculated over the middle $\frac{1}{3}$ of the spectrum. The critical value ${\lambda }_c$ at which the MBL transition happens is taken to be $3t$ and $3.6t$ for the AA and random models, respectively. We find the best scaling collapse occurs for $x\approx 0.4$ and $0.6$ for AA and random models, respectively. The dashed line is a guide to the eye for the data collapse. Note that the data for the AA model do not collapse well using the ansatz in Eq. (5). Inset (i): zoom-in version of the data collapse. We also collapsed the data for $L=14$ and 12 (not shown). For the AA model, we obtained the best data collapse exponent $x$ to be $\approx 0.6$ and $\approx 0.7$ for $L=14$ and 12, respectively, while for the random model, we obtained $x\approx 0.53$ and $0.62$ for $L=14$ and 12, respectively. As shown in inset (ii), the value of $x$ for the AA model decreases with increasing $L$ while the value of $x$ for the random model is roughly independent of $L$. Since only the data for the random model collapse well, we take $x=\nu \psi$ and use the estimate of $\nu \approx 0.7$ for the random model (see Appendix pp3) to get the value of $\psi$. We find the value of $\psi \approx 0.9$ for the random model.

Figure 11

Schematic crossover diagram for quasiperiodic systems with a finite system size $L$ versus the incommensurate potential strength $\lambda$. The vertical axis is plotted as $1/L^{1/\nu }$ (where $\nu$ is the correlation length exponent) so that the crossover lines separating each respective regime that are dictated by the correlation length $\xi$ are straight lines. We have included the dynamic exponent $z$ across the crossover diagram. Based on our numerical data, we argue that the finite-size quantum critical crossover regime is subdiffusive. In the thermodynamic limit, our data are consistent with a transition from a thermal diffusive phase to an MBL phase, different from the case of random systems.

Figure 12

Top panel: plot of optical conductivity $\sigma \left(\omega \right)$ vs $\omega$ of (a) AA and (b) random models with system size $L=16$ for various width $\eta$ of the Lorentzian function approximating the delta function in the Kubo formula [Eq. (2)]. The conductivity $\sigma \left(\omega \right)$ is calculated from $\frac{1}{3}$ of the full spectrum closest to zero energy. Bottom panel: dc conductivity $\left({\sigma }_{\mathrm{dc}}\right)$ vs $\eta$ of (c) AA and (d) random models for various values of disorder strength $\lambda$ and system size $L$.

Figure 13

Top panel: plot of optical conductivity $\sigma \left(\omega \right)$ vs $\omega$ of AA (left panel) and random (right panel) models with system size $L=16$ for different regions: thermal (top), near transition (middle), and MBL (bottom). The blue (red) curve corresponds to the optical conductivity averaged over samples with ${\sigma }_{\mathrm{dc}}<{\sigma }_{\mathrm{dc}}^{\mathrm{median}}$ $\left({\sigma }_{\mathrm{dc}}>{\sigma }_{\mathrm{dc}}^{\mathrm{median}}\right)$ where ${\sigma }_{\mathrm{dc}}^{\mathrm{median}}$ is the median value of the dc conductivity. The low-frequency tail is fitted to the formula $\sigma \left(\omega \right)\sim {\omega }^{\alpha }$ (dashed line) where ${\alpha }_{\mathrm{small}}$ $\left({\alpha }_{\mathrm{large}}\right)$ corresponds to the power-law exponent $\alpha$ of $\sigma \left(\omega \right)$ with ${\sigma }_{\mathrm{dc}}<{\sigma }_{\mathrm{dc}}^{\mathrm{median}}$ $\left({\sigma }_{\mathrm{dc}}>{\sigma }_{\mathrm{dc}}^{\mathrm{median}}\right)$. The conductivity $\sigma \left(\omega \right)$ is calculated over $\frac{1}{3}$ of the full spectrum closest to zero energy.

Figure 14

Finite-size critical scaling collapse for the level statistics (left panel) and entanglement entropy (right panel) for the AA model with obc (top panel), AA model with pbc (middle panel), and random model with pbc (bottom panel). The value of $\nu$ is $\sim 0.6$ for AA and $\sim 0.7$ for random model. The critical disorder strength ${\lambda }_c$ at which the MBL transition happens is larger for the random model.

Figure 15

Standard deviation of half-chain entanglement entropy over different samples $\left({\mathrm{\Delta }}_S^{\mathrm{samples}}\right)$, eigenstates $\left({\mathrm{\Delta }}_S^{\mathrm{states}}\right)$, and cuts $\left({\mathrm{\Delta }}_S^{\mathrm{cuts}}\right)$ for AA (left panel) and random (right panel) models with different system size $L$. The standard deviation is calculated over 100 states closest to the zero energy.