Hidden chiral symmetry for kagome lattice and its analogs

TL;DR

This paper introduces a novel form of chiral symmetry in non-bipartite lattice models with flat bands, using a molecular-orbital representation to analyze kagome lattices and their analogs, revealing topological edge modes.

Contribution

It develops a molecular-orbital framework to demonstrate chiral symmetry in non-bipartite flat-band models, extending understanding beyond traditional bipartite lattice systems.

Findings

Chiral symmetry can be preserved in non-bipartite lattices with flat bands using molecular orbitals.

The Pythagoras relation links the chiral operator and Hamiltonian in these models.

Topological edge modes emerge due to the chiral symmetry in kagome and related lattices.

Abstract

Chiral symmetry plays an essential role in condensed matter physics. In tight-binding models, it is often attributed to bipartite lattice structures, and its typical consequence is the ``particle-hole symmetric" band structures, that is, the positive and negative eigenenergies appear in a pairwise manner. In this work, we address the chiral symmetry for non-bipartite lattice models with flat bands. Our argument relies on what we call the molecular-orbital representation, by which we can guarantee the existence of the flat band. We show that the chiral symmetry is preserved for the non-flat bands when the molecular orbitals are normalized and divided into two sets within which they are non-overlapping. The chiral operator is constructed by the same molecular orbitals as the Hamiltonian. This results in the characteristic relation between the chiral operator and the Hamiltonian, which we call the Pythagoras relation. We present the examples of such models, i.e., the kagome lattice and its one-dimensional analog, and address the characteristics originating from the chiral symmetry, such as the emergence of the topological edge modes. We also show an example where the numbers of molecular orbitals in each set are different from each other, resulting in multiple flat bands with different energies.