# Topological nodal points in two coupled SSH chains

## TL;DR

This paper investigates the topological properties of two coupled SSH chains, revealing rich quantum phases, topological invariants, and phase transitions driven by nodal points and symmetry considerations.

## Contribution

It introduces a detailed analysis of topological nodal points and phase transitions in coupled SSH chains, highlighting symmetry-protected invariants and their role in quantum phase behavior.

## Key findings

- Identification of two topological insulating phases with edge states
- Topological nodal points characterized by kinks in a vector field in k space
- Second order quantum phase transition associated with nodal point creation

## Abstract

We study two coupled Su-Schrieffer-Heeger (SSH) chains system, which is shown to contain rich quantum phases associated with topological invariants protected by symmetries. In the weak coupling region, the system supports two non-trivial topological insulating phases, characterized by winding number N = +1 or -1, and two types of edge states. The boundary between the two topological phases arises from two band closing points, which exhibit topological characteristics in one-dimensional k space. By mapping Bloch states on a vector field in k space, the band degenerate points correspond to a pair of kinks of the field, with opposite topological charges. Two topological nodal points move and merge as the inter-chain coupling strength varies. This topological invariant is protected by the translational and inversion symmetries, rather than the antiunitary operation. Furthermore, we find that when a pair of nodal points is created, a second order quantum phase transition (QPT) occurs, associating with a gap closing and spontaneously symmetry breaking. This simple model demonstrates several central concepts in the field of quantum materials and provides a theoretical connection between them.