Zero Energy States for Commensurate Hopping Modulation of a Generalized Su-Schrieffer-Heeger Chain in the Presence of a Domain Wall

Abstract: We study the effect of domain wall (DW) on zero-energy states (ZESs) in the Su-Schrieffer-Heeger (SSH) chain. The chain features two fractional ZESs in the presence of such DW, one of which is localized at the edge and the other bound at the location of DW. This zero-energy DW state exhibits interesting modifications when hopping modulation is tuned periodically. We studied the energy spectra for commensurate frequencies $\theta=\pi,\pi/2,\pi/3$ and $\pi/4$. Following the recent study by the author of this paper [S. Mandal, S. Kar, Phys. Rev. B 109, 195124 (2024)], we showed numerically, along with physical intuition, that one ZES can bound at the DW position only for commensurate frequency $\theta=\frac{\pi}{2s+1}$ for zero or an integer $s$ values, while for $\theta=\frac{\pi}{2s}$ with nonzero or an integer $s$ value they appear only at the edges of the chain. We verify our numerical results by using exact analytical techniques. Both analyses indicate the realization of the Jackiw-Rebbi modes for our model only with $\theta=\frac{\pi}{2s+1}$. Moreover, the localization of zero-energy edge and DW states are investigated which reveals their localized (extended) nature for smaller (larger) $\Delta_{0}$ (amplitude of DW). The localization of topological DW states is suppressed as the width of DW ($\xi$) increases (typically scaled as $\sim 1/\xi$) while the edge state shows an extended behavior only for the large $\xi$ limit.