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

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, $E_c=\pm|2t-\lambda|,$ where $t$ is the hopping amplitude along both the legs and $\lambda$ 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 $0 < \nu <1$, where localized and weakly delocalized states coexist even beyond the transition point ($\lambda_c=2t$). 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.