Parity-dependent reentrant topology in a Su--Schrieffer--Heeger chain with power-law quasiperiodic modulation

We investigate reentrant topological transitions in a one-dimensional Su–Schrieffer–Heeger chain with power-law quasiperiodically modulated intracell hopping. The modulation is characterized by a positive integer exponent $n$ and a tunable parameter $\beta$, which continuously interpolates between the smooth power-law quasiperiodic limit and a sign-function limit that becomes square-wave-like for odd $n$ and uniform for even $n$. By combining analytical calculations of the zero-mode inverse localization length with numerical evaluations of a real-space topological indicator, we determine the topological phase diagrams in the $\beta \to 0$, $\beta \to \mathrm{\infty }$, and finite-$\beta$ regimes. We show that deterministic quasiperiodic modulation can induce TAI-like reentrant topological phases within finite parameter windows. The formation of these phases depends crucially on the parity of $n$: for positive modulation strength, odd-power modulations can induce reentrant topology from the clean trivial regime $\mathrm{\mid }{t}_1\mathrm{\mid }>1$, whereas even-power modulations allow such reentrance only from the negative clean trivial regime $t_1<-1$. Exact analytical expressions for the zero-mode inverse localization length are obtained for $n=1,2,3,4$, yielding explicit or implicit transition conditions. The finite-$\beta$ results demonstrate that the parity-dependent structure remains robust throughout the interpolation between the two limiting cases. This parity effect originates from whether the modulation preserves or removes the sign structure of $\cos x$. We further propose an electrical-circuit implementation and discuss experimentally accessible signatures of the reentrant trivial–topological–trivial transition.