Many-body localization in the interpolating Aubry-André-Fibonacci model

We investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbor coupling, subject to an external quasiperiodic on-site magnetic field. The quasiperiodic modulation interpolates between two paradigmatic models, namely the Aubry-André and the Fibonacci models. We find that stronger many-body interactions extend the ergodic phase in the former, whereas they shrink it in the latter. Furthermore, the many-body localization transition points at the two limits of the interpolation appear to be continuously connected along the deformation of the quasiperiodic modulation. As a result, the position of the many-body localization transition depends on the interaction strength for an intermediate degree of deformation. Moreover, in the region of parameter space where the single-particle spectrum contains both localized and extended states, many-body interactions induce an anomalous effect: weak interactions localize the system, whereas stronger interactions enhance ergodicity. We map the model's localization phase diagram using the decay of the quenched spin imbalance in relatively long chains. This is accomplished employing a time-dependent variational approach applied to a matrix product state decomposition of the many-body state. Our model serves as a rich playground for testing many-body localization under tunable potentials.

Figure 1

i-IAAF model and its many-body localization properties. (a) Quasiperiodic IAAF potential as a function of space [cf. Eq. (2)] for different values of $\beta$. The continuous function is plotted and its discrete sampling is marked by dots. (b) Sketch of the many-body localization phase diagrams for the i-IAAF [cf. Eq. (3)], established in this work for $\mathrm{\Delta }=1$ and $\mathrm{\Delta }=0.3$. The nonergodic MBL phase and the delocalized ergodic phase are marked in red and blue, respectively.

Figure 2

Localization properties of the noninteracting IAAF model; cf. Eq. (3) with $\mathrm{\Delta }=0$. (a) The inverse participation ratio (IPR) for every single-particle eigenstate in the spectrum as a function of $\beta$ at constant $\lambda /t=8$. The dashed oval marks the region where the lowest band bundle of eigenstates becomes extended. (b) Localization phase diagram of the ground state obtained by the IPR described in the main text. (c) Localization phase diagram obtained by averaging the IPR over the lowest band bundle. For all plots, we used a system with $L=144$ sites.

Figure 3

Numerical simulation of the MBL transition in the Fibonacci model. We show the time evolution of the averaged imbalance (5) using TDVP for three amplitudes, $\lambda /t$, of the quasiperiodic potential and two interaction strengths (a) $\mathrm{\Delta }=1$ and (b) $\mathrm{\Delta }=0.3$. In both cases, for $\lambda /t=2.2$ the averaged imbalance decays (red line), indicating that this point lies in the ergodic phase. For the other two presented plots (blue and green lines), $\overline{\mathcal{I}}$ saturates over time to a constant. (c) The inverse power law coefficient (5) fitted to $\overline{\mathcal{I}\left(\tau \right)}$ over a finite fitting time window $\tau \in [50,200]$ [dashed lines in (a) and (b)] as a function of $\lambda /t$. For all plots we use $L=50$ sites and averaged the imbalance over 36 realizations with different $\varphi$ taken from the interval $\varphi \in [0,2\pi ]$. The bond dimension used for all plots is $\chi =64$ and the time step is $\delta \tau =0.1$. Error bars are $1\sigma$ intervals based on a bootstrapping procedure [70].

Figure 4

Numerical simulation of the MBL transition in the i-IAAF model. (a) Many-body localization phase diagram obtained from the exponent $\gamma$ [cf. Eq. (5)], for (left) strong interaction, $\mathrm{\Delta }=1$, and (right) weak interaction, $\mathrm{\Delta }=0.3$. The bottom panels show $\gamma$ as a function of $\beta$ for selected values of $\lambda /t$ for (b) $\mathrm{\Delta }=1$ and (c) $\mathrm{\Delta }=0.3$. The numerical specifications used are the same as in Fig. 3, with the distinction that the fitting time window now depends on $\lambda /t$ as described in Appendix pp2.

Figure 5

Interaction-dependent localization-delocalization transition. (a) Sketch of the overlapped interacting ($\mathrm{\Delta }=1$) and noninteracting ($\mathrm{\Delta }=0$) localization phase diagrams; cf. Figs. 2 and 4. Solid lines denote the transition from extended to localized phase in the noninteracting limit, $\mathrm{\Delta }=0$; see also Fig. 2. Dashed line denotes the MBL transition for $\mathrm{\Delta }=1$. (b) Time evolution of the averaged imbalance (5) at a constant $\lambda /t=4$ and $\beta =1.4$ [red point in (a)]. (c) The exponent of the averaged imbalance decay (5), as a function of interaction strength $\mathrm{\Delta }$ for several points in the phase diagram [also marked in (a)]. We pick points that lie within the delocalized-phase sliver of the noninteracting model, i.e., situated at finite $\beta \sim 2$ and at $\lambda /t>2$. System size in all plots is $L=50$, the number of different realizations of the quasiperiodic potential is 36, and the inverse power law is fitted in a time window [10, 200]. The bond dimension used for all plots is $\chi =64$ and the time step is $\delta \tau =0.1$. Error bars are $1\sigma$ intervals based on a bootstrapping procedure.

Figure 6

(a) Time evolution of the imbalance (5) for two different values of interaction at the constant $\lambda /t=3.8$ in the Fibonacci limit; cf. Fig. 3. (b) Fitted power-law exponent as a function of the potential strength for the strong and weak interaction cases. We use the following parameters: $L=16, \chi =256$, and $\delta \tau =0.1$. Error bars are obtained from $1\sigma$ intervals based on a bootstrapping procedure.

Figure 7

Power-law exponent $\gamma$ as a function of $\beta$ for two values of a bond dimension $\chi$ and at several values of $\lambda /t$. Panels (a) and (b) show the results for strong interaction, $\mathrm{\Delta }=1$, at $\lambda /t=3$ and $\lambda /t=4$, respectively. Dependencies in panels (c) and (d) are calculated for a weak interaction, $\mathrm{\Delta }=0.3$, and at $\lambda /t=3.5$ and $\lambda /t=5$, respectively. The length of the system is $L=50$ sites and the time step used for all plots is $\delta \tau =0.1$.

Figure 8

Imbalance oscillations in the Fibonacci model. (a) Sketch of the mechanism responsible for strong oscillations in the Fibonacci limit. A particle, initially placed in a well formed by two “A” sites (with the lower potential $-\lambda$) and bounded by “B” sites (with the higher potential $+\lambda$), hops inside the well, giving rise to the oscillations of the imbalance; see Figs. 3 and 3. (b) Comparison between the numerical result for $\lambda /t=20$ and Eq. (C7). For the numerical calculation, we used $L=50, \chi =64$, and $\delta \tau =0.1$.

Figure 9

Mean gap ratio $\overline{r}$ as a function of $\lambda /t$ for several different values of $\beta$. We used a system with strong interaction $\mathrm{\Delta }=1$ and $L=14$.