The exact mobility edges in SVQL (slowly-varying quasiperiodic ladder) model

TL;DR

The paper addresses mobility edges in a minimal quasiperiodic ladder system by introducing a slowly varying inter-leg (bond) modulation without onsite quasiperiodicity. By mapping the two-leg ladder to symmetric and antisymmetric channels in the adiabatic limit, the authors derive an exact mobility-edge condition $E_c=\pm|2t-\lambda|$ and show that the two channels experience opposite effective onsite potentials, enabling energy-dependent coexistence of localized and extended states. Numerical diagnostics using IPR, Lyapunov exponents, DOS, and PR scaling confirm the ME across a range of $\nu$ and $\lambda$, revealing a nonergodic metallic-like phase; as $\nu\to 1$ the model connects to a rung-modulated Aubry–André–Harper-like behavior with a sharp transition at $\lambda_c=2t$. The work provides a robust bond-modulation mechanism for MEs in ladder systems, with experimental implications for cold atoms and photonic lattices and potential avenues for studying many-body localization and higher-dimensional generalizations.

Abstract

We propose a minimal two-leg ladder model in which the mobility edge (ME) arises solely due to bond modulation, introduced through a slowly varying quasiperiodic modulation in the inter-leg tunnelling amplitudes. We demonstrate that this bond-modulated ladder naturally hosts two propagation channels, whose symmetric and antisymmetric combinations experience opposite effective onsite potentials, unlike the one-dimensional quasiperiodic models with onsite modulations. Using the adiabatic (slowly varying) limit of the modulation, we derive an exact analytical condition for the single-particle mobility edge, $E_c=\pm|2t-λ|,$ where $t$ is the hopping amplitude along both the legs and $λ$ is the bond modulation strength. This result directly generalizes the classic ME condition for slowly varying onsite potentials to a multi-leg (two-leg in our case) geometry. Extensive numerical calculations, including inverse participation ratios, Lyapunov exponents, density of states, and participation-ratio scaling, demonstrate excellent agreement with the analytical prediction across a wide range of parameters. We further identify a regime for small modulation exponents $0 < ν<1$, where localized and weakly delocalized states coexist even beyond the transition point ($λ_c=2t$). Our results establish that a deterministic bond modulation can serve as a sufficient ingredient to produce an exact ME in ladder systems, offering experimentally accessible routes toward tuning nonergodic extended phases.

Figures (9)

• Figure 1: Schematic representation of the Slowly Varying Quasiperiodic Ladder (SVQL) model. The system consists of two parallel tight-binding chains (legs), coupled through inter-leg rung hopping amplitudes $\Gamma_n= \lambda \cos{(2Qn^{\nu})}$, which incorporate a deterministic slowly varying quasiperiodic modulation. Horizontal links represent uniform intra-leg hopping t, which is the same in both legs.
• Figure 2: A heatmap of average IPR for the SVQL model in the ($\nu-\lambda$) parameter space. The diagram displays the emergence of distinct spectral regimes: extended (blue) and localized (red).
• Figure 3: (a)-(c) are the contour plots of $IPR$ as function of $\lambda$ and energy($E$) and (d)-(f) are the contour plots of $IPR$ as function of $\lambda$ and state index ratio($\frac{k}{L}$, where $k$ is the state index and $L$ is the total number of site) for all eigenstates. Calculations are performed for the total system size (L) = 2000 and for $\nu$ = 0.7, 0.75, and 0.8, respectively; $t$ is fixed to 1 for the calculations. The plots reveal clear energy-dependent MEs whose position is $\nu$ independent, confirming that slower quasiperiodicity enhances localization contrast and sharpens the extended-localized spectral separation.
• Figure 4: (a)-(c) are the plots of inverse localization length($\gamma$) as a function of energy eigenvalues ($E$) for $\nu=0.7, 0.75, 0.8$ respectively for $\lambda=0.4t$. Extended states correspond to $\gamma(E) \approx 0$, while localized states appear as finite $\gamma(E)$. Increasing $\nu$ enhances the sharpness of the ME, indicating stronger energy-selective localization in the SVQL model. For $\lambda=0.4t$, all the middle states are delocalized, and the ME appears at $E_c=-1.6t$ and $+1.6t$ and $t$ is fixed to 1. We take $L=2000$ for numerics.
• Figure 5: (a)-(c) represents the inverse localization length($\gamma$) as a function of eigen energy($E$) for $\nu=0.7,0.75, 0.8$ values respectively for $\lambda=2t$. In this case, the ME arises at $E_c=0,$ consistent with the derived expression $E_c=\pm|2t-\lambda|.$ We take $L=2000$ and $t=1$ for numerics.