Even--odd classification of reentrant topology in a Su--Schrieffer--Heeger chain with integer-power quasiperiodic hopping
We investigate reentrant topology in a one-dimensional Su--Schrieffer--Heeger chain with integer-power quasiperiodically modulated intracell hopping. The modulation is controlled by a positive integer exponent $n$ and a tunable parameter $β$, which interpolates between a smooth integer-power quasiperiodic profile and a sign-function limit. Combining the zero-mode inverse-localization-length criterion with a real-space topological indicator, we determine the phase diagrams in the $β\to0$, $β\to\infty$, and finite-$β$ regimes. We find an even--odd classification of reentrant topological windows governed by the dc component and support of the full modulation profile. For positive modulation strength, odd powers yield a zero-mean sign-changing profile and can induce reentrance from the clean trivial regime $|t_1|>1$, whereas even powers yield a positive-mean non-negative profile and allow reentrance only from the negative clean trivial regime $t_1<-1$. Although even-power profiles contain sign-changing fluctuations after subtracting their positive average, the full modulation profile remains non-negative. This even--odd structure of the full modulation profile provides a control knob for topological Anderson-like reentrant phases. We further discuss the associated bulk localization properties and show that the phase diagrams are robust against moderate hopping fluctuations, suggesting a feasible realization in electrical circuits.