Extended states in one-dimensional aperiodic lattices with linearly varying patches

We introduce a family of one-dimensional aperiodic tight-binding models with linearly varying patches of $A$-type sites with on-site energies ${\ensuremath{\epsilon}}_A=0$ connected by single $B$-type sites with ${\ensuremath{\epsilon}}_B=W$. We analytically show such structures have strong spatial correlations. We theoretically find states are extended at resonance levels in the vicinity of ${E}_M^{\ensuremath{\kappa}}=\ensuremath{-}2cos\frac{\ensuremath{\kappa}\ensuremath{\pi}}{M}$ if they are allowed energies, where $M=md$ are the size differences of patches, $d$ is the variation rate of patch sizes, $m\ensuremath{\in}{\mathcal{N}}₊$, and $\ensuremath{\kappa}=1,2,...,M\ensuremath{-}1$. Related delocalization-localization transitions are explored. Numerical evidence is in excellent quantitative agreement with theoretical predictions.