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arXiv:2505.17876 · Physical review. B./Physical review. B

Subsystem localization in a two-leg ladder system

Arpita Goswami, Pallabi Chatterjee, Ranjan Modak, Shaon Sahoo

Published Oct 16, 2025• 3 citations
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Abstract

We consider a ladder system where one leg, referred to as the ``bath,'' is governed by an Aubry-Andr\'e (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 (${t}^{\ensuremath{'}}>{t}_{c}^{\ensuremath{'}}$), 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 wave-packet 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 ${t}^{\ensuremath{'}}<{t}_{c}^{\ensuremath{'}}$, 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 ${t}^{\ensuremath{'}}/V\ensuremath{\ll}1$, which continuously approaches the ballistic behavior as ${t}^{\ensuremath{'}}\ensuremath{\rightarrow}0$. Finally, in some limiting scenario, we also establish a mapping between our ladder system and a well-studied one-dimensional generalized Aubry-Andr\'e (GAA) model.

DOI
10.1103/95t3-6qd4

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We consider a ladder system where one leg, referred to as the bath,'' is governed by an Aubry-Andr\'e (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\ensuremath′>tc\ensuremath′{t}^{\ensuremath{'}}>{t}_{c}^{\ensuremath{'}}t\ensuremath′>tc\ensuremath′​), the analysis of the static properties shows that there are three distinct phases as the AA potential strength VVV is varied: a fully delocalized phase at low VVV, a localized phase at intermediate VVV, and a weakly delocalized (fractal) phase at large VVV. The fractal phase also appears in a narrow region along the boundary between the delocalized and localized phases. An analysis of the projected wave-packet 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\ensuremath′<tc\ensuremath′{t}^{\ensuremath{'}}<{t}_{c}^{\ensuremath{'}}t\ensuremath′<tc\ensuremath′​, the intermediate localized phase disappears and we find a delocalized (ballistic) phase at low VVV and a weakly delocalized (subdiffusive) phase at large VVV. 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 VVV, we also identify a superdiffusive regime emerging in the limit t\ensuremath′/V\ensuremath≪1{t}^{\ensuremath{'}}/V\ensuremath{\ll}1t\ensuremath′/V\ensuremath≪1, which continuously approaches the ballistic behavior as t\ensuremath′\ensuremath→0{t}^{\ensuremath{'}}\ensuremath{\rightarrow}0t\ensuremath′\ensuremath→0. Finally, in some limiting scenario, we also establish a mapping between our ladder system and a well-studied one-dimensional generalized Aubry-Andr'e (GAA) model.
Periodic driving provides a powerful route for controlling localization and engineering effective Hamiltonians in quantum systems. A localized one-dimensional Aubry-André chain can be driven into a fully delocalized phase when the entire system is driven below a critical frequency. Here, we ask whether such delocalization can be achieved by driving only a part of the system. We address this question using a two-leg ladder comprising an Aubry-André chain coupled to a periodically driven clean tight-binding chain. Exact numerical calculations show that, starting from a localized static ladder, subsystem driving below a critical frequency cannot achieve complete delocalization. Instead, a mixed phase emerges, with localized and delocalized states coexisting and the fraction of delocalized states varying nonmonotonically with the driving frequency. This behavior can be understood using local effective Hamiltonians. Although these effective Hamiltonians are derived analytically in the high-frequency regime, they can nevertheless predict the localization transition occurring in the moderate-frequency regime. In fact, by quantifying the emergent nonlocality of the Floquet Hamiltonian, we show that these effective Hamiltonians provide a valid description up to a crossover frequency that is lower than the critical frequency associated with the localization transition. Our results demonstrate an alternative route to Floquet engineering, in which the transport properties of a quantum lattice system can be modified by driving only a subsystem.
We study quantum transport in a quasiperiodic Aubry-André-Harper (AAH) model induced by the coupling of the system to a Markovian heat bath. We find that coupling the heat bath locally does not affect transport in the delocalized and critical phases, while it induces logarithmic transport in the localized phase. Increasing the number of coupled sites at the central region introduces a transient diffusive regime, which crosses over to logarithmic transport in the localized phase and in the delocalized regime to ballistic transport. On the other hand, when the heat bath is coupled to equally spaced sites of the system, we observe a crossover from ballistic and logarithmic transport to diffusion in the delocalized and localized regimes, respectively. We propose a classical master equation, which in the localized phase, captures our numerical observations for both coupling configurations on a qualitative level and for some parameters, even on a quantitative level. Using the classical picture, we show that the crossover to diffusion occurs at a time that increases exponentially with the spacing between the coupled sites, and the resulting diffusion constant decreases exponentially with the spacing.
The critical phase of a non-Hermitian quasicrystal can support stronger transport than its surrounding delocalized phase. We demonstrate this anomalous behavior in the one-dimensional nonreciprocal Aubry-André-Harper model through a combined study of dynamical quantum phase transitions (DQPTs) and wavepacket diffusion. Using a parity-sorted energy-spectrum classification that directly encodes the generalized PT\mathcal{PT}PT symmetry, we find that DQPTs in this system are energy-resolved, in contrast to the energy-independent DQPTs of Hermitian quasicrystals. The energy-resolved features are most pronounced when the initial and final Hamiltonians belong to different phases (localized or extended), and they are tied to the even-odd index structure of the spectrum, which we exploit to organize the quench-dynamical landscape. For wavepacket dynamics after a single-site quench, the diffusion exponent βββ, extracted from the long-time power-law scaling of the root-mean-square displacement σ(τ)σ(τ)σ(τ), partitions the phase diagram into four distinct regimes. In the Hermitian limit the extended phase is ballistic (β=1β=1β=1), the critical phase is normally diffusive (β=0.5β=0.5β=0.5), and the localized phase yields β→0β\to 0β→0. Nonreciprocity reverses this hierarchy: the extended phase becomes normally diffusive, while the critical phase turns ballistic. We trace the anomalous β=1β=1β=1 at criticality to the self-similar multifractal structure of the critical eigenstates, whose nodal positions are organized by the golden ratio. A finite-size scaling ansatz built on the wave-front propagation yields σ(τ)∝τσ(τ)\proptoτσ(τ)∝τ. The parity-resolved DQPTs and the βββ-phase diagram establish two complementary dynamical diagnostics of nonreciprocal quasicrystals, in which nonreciprocity promotes transport at the critical point and suppresses it in the delocalized phase.
We demonstrate proximity-induced localization and re-entrant delocalization in a detuned quasiperiodic ladder. In the weak-quasiperiodic regime, where uncoupled Aubry-André chains are extended, a staggered detuning applied to only one leg induces localization in the neighboring leg through inter-leg hybridization. With further increasing detuning, the pure chain re-enters a delocalized regime, producing a delocalized-localized-delocalized sequence without directly modifying its quasiperiodic potential. We identify these regimes using static localization diagnostics and demonstrate experimentally accessible dynamical signatures on IBM Quantum hardware: strongly suppressed wavefunction spreading in the proximity-localized regime and its re-entrant recovery at larger detuning. These results establish leg-selective detuning as a programmable control knob for inducing and reversing localization in coupled quasiperiodic systems.
Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions D q > 0 . In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions D 2 and D ∞ , relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of nonergodic phases in the entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
We numerically study spectral and wave-function properties, as well as time evolution, in finite-size commensurate approximations of the 3D Aubry-André model. We explicitly show, through scaling analysis, the existence of two critical points. One of the critical points is a localization-delocalization transition in real-space, similar to the critical point in the 3D Anderson model. The other is a localization-delocalization transition in a dual momentum space, which does not have any analog in the 3D Anderson model. We estimate their values from spectral measures, such as the gap ratio and the spectral form factor, as well as from wave-function measures, such as the inverse participation ratio and the participation entropy. Additionally, we demonstrate that both critical points exhibit hallmarks of critical points between quantum ergodicity and localization. First, the eigenstates and the eigenspectrum at the critical point fulfill the mutual universal relation between the eigenspectrum compressibility χχχ and the eigenstate fractal dimension d(1)d^{(1)}d(1) [S. Jiricek et al., Proc. Nat. Acad. Sci. 123, e2518027123 (2026)]. Second, the dynamics of the wave packet at the critical point is governed by the critical dynamical exponent zzz, characterized by the spectral fractal dimension and the dimensionality of the lattice [M. Hopjan and L. Vidmar, Phys. Rev. Lett. 135, 060401 (2025)]. Thus, similarly to the 3D Anderson model, the critical dynamical exponent is sub-diffusive with the value of z=3z=3z=3.