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

TL;DR

This paper investigates how domain walls affect zero-energy states in a generalized SSH chain with periodic hopping modulation, revealing conditions for Jackiw-Rebbi modes and analyzing their localization properties.

Contribution

The study provides analytical and numerical insights into the conditions under which zero-energy states localize at domain walls or edges in a modulated SSH chain, extending understanding of topological states.

Findings

Zero-energy states localize at domain walls only for specific commensurate frequencies.

Jackiw-Rebbi modes are realized only at certain modulation frequencies.

Localization of states depends on the domain wall width and amplitude.

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.