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arXiv:2204.02717

Uniform Anderson Localization in One-Dimensional Floquet Maps

I. Vakulchyk, Sergej Flach A

Published Apr 6, 2022• 0 citations
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Abstract

We study Anderson localization in a discrete-time quantum map dynamics in one dimension with nearest-neighbor hopping strength $θ$ and quasienergies located on the unit circle. We demonstrate that strong disorder in a local phase field yields a uniform spectrum gaplessly occupying the entire unit circle. The resulting eigenstates are exponentially localized. Remarkably this Anderson localization is universal as all eigenstates have one and the same localization length $L_{loc}$. We present an exact theory for the calculation of the localization length as a function of the hopping, $1/L_\text{loc}=\left|\ln\left(|\sin(θ)|\right)\right|$, that is tunable between zero and infinity by variation of the hopping $θ$.

DOI
10.48550/arxiv.2204.02717

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We study Anderson localization in a discrete-time quantum map dynamics in one dimension with nearest-neighbor hopping strength θθθ and quasienergies located on the unit circle. We demonstrate that strong disorder in a local phase field yields a uniform spectrum gaplessly occupying the entire unit circle. The resulting eigenstates are exponentially localized. Remarkably this Anderson localization is universal as all eigenstates have one and the same localization length LlocL_{loc}Lloc​. We present an exact theory for the calculation of the localization length as a function of the hopping, 1/Lloc=∣ln⁡(∣sin⁡(θ)∣)∣1/L_\text{loc}=\left|\ln\left(|\sin(θ)|\right)\right|1/Lloc​=∣ln(∣sin(θ)∣)∣, that is tunable between zero and infinity by variation of the hopping θθθ.
We study the interplay of two interacting discrete time quantum walks in the presence of disorder. Each walk is described by a Floquet unitary map defined on a chain of two-level systems. Strong disorder induces a novel Anderson localization phase with a gapless Floquet spectrum and one unique localization length ${\ensuremath{\xi}}_{1}$ for all eigenstates for noninteracting walks. We add a local contact interaction which is parametrized by a phase shift $\ensuremath{\gamma}$. A wave packet is spreading subdiffusively beyond the bounds set by ${\ensuremath{\xi}}_{1}$ and saturates at a new length scale ${\ensuremath{\xi}}_{2}\ensuremath{\gg}{\ensuremath{\xi}}_{1}$. In particular we find ${\ensuremath{\xi}}_{2}\ensuremath{\sim}{\ensuremath{\xi}}_{1}^{1.2}$ for $\ensuremath{\gamma}=\ensuremath{\pi}$. We observe a nontrivial dependence of ${\ensuremath{\xi}}_{2}$ on $\ensuremath{\gamma}$, with a maximum value observed for $\ensuremath{\gamma}$ values which are shifted away from the expected strongest interaction case $\ensuremath{\gamma}=\ensuremath{\pi}$. The novel Anderson localization regime indicates violation of single parameter scaling for both interacting and noninteracting walks.
We study Anderson localization in a generalized discrete-time quantum walk---a unitary map related to a Floquet driven quantum lattice. It is controlled by a quantum coin matrix which depends on four angles with the meaning of potential and kinetic energy, and external and internal synthetic flux. Such quantum coins can be engineered with microwave pulses in qubit chains. The ordered case yields a two-band eigenvalue structure on the unit circle, which becomes completely flat in the limit of vanishing kinetic energy. Disorder in the external magnetic field does not impact localization. Disorder in all the remaining angles yields Anderson localization. In particular, kinetic-energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. Strong disorder in potential and internal magnetic field energies allows one to obtain analytical expressions for spectrally independent localization length, which is highly useful for various applications.
The random hopping models exhibit many fascinating features, such as diverging localization length and density of states as energy approaches the band center due to its particle-hole symmetry. Nevertheless, such models are yet to be realized experimentally because the particle-hole symmetry is easily destroyed by diagonal disorder. Here we propose that a pure random hopping model can be effectively realized in ultracold atoms by modulating a disordered onsite potential in particular frequency ranges. This idea is motivated by the recent development of the phenomena called ``dynamical localization'' or ``coherent destruction of tunneling.'' Investigating the application of this idea in one dimension, we find that if the oscillation frequency of the disorder potential is gradually increased from zero to infinity, one can tune a noninteracting system from an Anderson insulator to a random hopping model with diverging localization length at the band center, and eventually to a uniform-hopping tight-binding model.
We investigate the effects of disorder on a periodically driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady-state nearly quantized current. Remarkably, this is linked to a localization-delocalization transition in the Floquet states at strong disorder, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.
The problem of Anderson localization, as well as the single particle localization-delocalizaton transition of the Aubry-André model, is studied employing operator Krylov space methods. It is shown that even when the dynamics is generated by a Hamiltonian, studying the dynamics at stroboscopic rather than continuous times has its advantages. In particular, mapping the dynamics to an effective Floquet problem results in an operator Krylov space description where quantities such as the spectral function can be computed with fewer computational resources, while a moment method exists that allows for the extraction of Krylov parameters directly from the discrete time autocorrelation function. For stroboscopic dynamics, the operator Krylov space corresponds to the dynamics of an edge operator of an inhomogeneous Floquet transverse field Ising model, with the parameters of this effective model generated recursively. The Krylov parameters show disorder-averaged renormalization with their distribution narrowing as the recursion step increases. It is shown that a more physical spectral function is obtained from the Krylov parameters obtained from the disorder-averaged autocorrelation function, rather than the disorder-averaged Krylov parameters. The delocalized (localized) phase is shown to correspond to the appearance (absence) of a Porter-Thomas distribution, a ballistically propagating (localized) wavefront in operator Krylov space, and a smooth (discrete) Berstein-Szegö power-spectrum. The localization-delocalization transition is also demonstrated in operator Krylov space. A Porter-Thomas distribution is also observed at the critical point. The long-time dynamics and the inverse participation ratio at the critical point is shown to exhibit behavior consistent with a multi-fractal scaling with system size.
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