Subsystem localization

TL;DR

We investigate a two-leg ladder where the bath leg hosts an Aubry–André-type quasi-periodic potential and the subsystem leg is a clean tight-binding chain. By tuning the inter-leg coupling $t'$ and the bath strength $V$, we map out a rich phase diagram featuring fully delocalized (ballistic), fully localized, and weakly delocalized (fractal) phases, along with an anomalous crossover region and a regime that becomes ballistic as $t'\to0$. The analysis combines static projections of full eigenstates to define $IPR_A$ and $\langle PR_A\rangle$, scaling exponents $\eta$, and dynamic exponents $\gamma$ from projected wavepacket dynamics, and is complemented by a minimal four-site toy model and an effective 1D mapping to a generalized Aubry–André chain. The results show that a localized bath can induce localization in the attached subsystem within a finite parameter window, while large bath potential can reintroduce delocalized/fractal transport, revealing controllable transport properties via bath parameters with potential relevance for cold-atom and quantum-device setups. A simple effective-model derivation and GAA-like mapping provide conceptual coherence for the observed ballistic, superdiffusive, and subdiffusive behaviors.

Abstract

We consider a ladder system where one leg, referred to as the ``bath", is governed by an Aubry-André (AA) type Hamiltonian, while the other leg, termed the ``subsystem", follows a standard tight-binding Hamiltonian. We investigate the localization properties in the subsystem induced by its coupling to the bath. For the coupling strength larger than a critical value ($t'>t'_c$), the analysis of the static properties shows that there are three distinct phases as the AA potential strength $V$ is varied: a fully delocalized phase at low $V$, a localized phase at intermediate $V$, and a weakly delocalized (fractal) phase at large $V$. The fractal phase also appears in a narrow region along the boundary between the delocalized and localized phases. An analysis of the projected wavepacket dynamics in the subsystem shows that the delocalized phase exhibits a ballistic behavior, whereas the weakly delocalized phase is subdiffusive. Interestingly, the narrow fractal phase shows a super- to subdiffusive behavior as we go from the delocalized to localized phase. When $t'<t'_c$, the intermediate localized phase disappears, and we find a delocalized (ballistic) phase at low $V$ and a weakly delocalized (subdiffusive) phase at large $V$. Between those two phases, there is also an anomalous crossover regime where the system can be super- or subdiffusive. Beyond the ballistic phase observed at low $V$, we also identify a superdiffusive regime emerging in the limit $t'/V \ll 1$, which continuously approaches the ballistic behavior as $t' \to 0$. Finally, in some limiting scenario, we also establish a mapping between our ladder system and a well-studied one-dimensional generalized Aubry-André (GAA) model.

Figures (24)

• Figure 1: Schematic diagram of the model: the subsystem ($A$) is described by a standard TB Hamiltonian whereas the bath ($B$) is governed by an AA type potential.
• Figure 2: Schematic phase diagram in rescaled parameter space of $\tilde{t'}=t'/t'_c$ and $\tilde{V}=V/t'_c$. P1 - delocalized (ballistic) phase, P2 - localized phase, P3 - weakly delocalized (fractal and subdiffusive) phase, P4 - an anomalous crossover regime (fractal and super- or sub-diffusive), and P5 - weakly delocalized (fractal and superdiffusive) phase.
• Figure 3: (a)-(c): $\nu$ vs. $V$ plots for $t'=$ 1, 5 and 10, respectively. The calculations are performed for the total subsystem size N = 500.
• Figure 4: Plots of $IPR_A$ as function of $V$ for all $\epsilon$-significant states ($\epsilon$ = 0.3, 0.5 and 0.7). Calculations are performed for the total subsystem size N = 500 and for $t'$ = 1, 5 and 10.
• Figure 5: Variation of $V_1$ and $V_2$ as function of $t'$. The extrapolation of data shows that $V_1\approx V_2$ at $t'=t'_c\approx 4.4$ (represented by a star symbol).