Quantum criticality and Kibble-Zurek scaling in the Aubry-Andr\'{e}-Stark model

We explore quantum criticality and Kibble-Zurek scaling (KZS) in the Aubry-Andre-Stark (AAS) model, where the Stark field of strength $\varepsilon$ is added onto the one-dimensional quasiperiodic lattice. We perform scaling analysis and numerical calculations of the localization length, inverse participation ratio (IPR), and energy gap between the ground and first excited states to characterize critical properties of the delocalization-localization transition. Remarkably, our scaling analysis shows that, near the critical point, the localization length $\xi$ scales with $\varepsilon$ as $\xi\propto\varepsilon^{-\nu}$ with $\nu\approx0.3$ a new critical exponent for the AAS model, which is different from the counterparts for both the pure Aubry-Andre (AA) model and Stark model. The IPR $\mathcal{I}$ scales as $\mathcal{I}\propto\varepsilon^{s}$ with the critical exponent $s\approx0.098$, which is also different from both two pure models. The energy gap $\Delta E$ scales as $\Delta E\propto \varepsilon^{\nu z}$ with the same critical exponent $z\approx2.374$ as that for the pure AA model. We further reveal hybrid scaling functions in the overlap between the critical regions of the Anderson and Stark localizations. Moreover, we investigate the driven dynamics of the localization transitions in the AAS model. By linearly changing the Stark (quasiperiodic) potential, we calculate the evolution of the localization length and the IPR, and study their dependence on the driving rate. We find that the driven dynamics from the ground state is well described by the KZS with the critical exponents obtained from the static scaling analysis. When both the Stark and quasiperiodic potentials are relevant, the KZS form includes the two scaling variables. This work extends our understanding of critical phenomena on localization transitions and generalizes the application of the KZS to hybrid models.