# Edge States of a Periodic Chain with Four-Band Energy Spectrum

## TL;DR

This paper analytically investigates edge states in a finite four-band tight-binding chain with alternating hopping parameters, revealing multiple topological phases distinguished by gauge-invariant indices, supported by numerical validation.

## Contribution

It introduces an analytical solution for edge states in a four-band chain with alternating hoppings and identifies new topological indices distinguishing different edge phases.

## Key findings

- Existence of four edge phases for odd sites
- Existence of eight edge phases for even sites
- Identification of gauge-invariant topological indices

## Abstract

Tight-binding model on a finite chain is studied with four-fold alternated hopping parameters $t_{1,2,3,4}$. Imposing the open boundary conditions, the corresponding recursion is solved analytically with special attention paid to the occurrence of edge states. Corresponding results are strongly corroborated by numeric calculations. It is shown that in the system there exist four different edge phases if the number of sites is odd, and eight edges phases if the chain comprises even number of sites. Phases are labelled by $\sigma_1\equiv{\rm sgn}(t_1t_3-t_2t_4)$, $\sigma_2\equiv{\rm sgn}(t_1t_4-t_2t_3)$ and $\sigma_3\equiv{\rm sgn}(t_1t_2-t_3t_4)$. It is shown that these quantities represent gauge invariant topological indices emerging in the corresponding infinite chains.