Chiral symmetry in condensed matter physics implies the sublattice symmetry (as far as the one-particle hopping Hamiltonian is concerned). Then the usual belief is that the nearest neighbor hopping Hamiltonians on a square lattice and a honeycomb lattice are chiral symmetric but not on a Kagome lattice. To be chiral symmetric, whole lattice points are divided into two classes (sublattices) and the hopping only connects between them not within themselves (bipartite lattice). Formally, it implies that the one-particle Hamiltonian,H, anticommutes with the chiral operator ɣ as {H,ɣ}=H ɣ+ɣ H=0. Energy bands of the square lattice and the honeycomb lattice are symmetric in energy (positive/negative), which is a direct outcome of the chiral symmetry.
In our recent paper, “Hidden chiral symmetry for the kagome lattice and its analogs” by Tomonari Mizoguchi and Yasuhiro Hatsugai, that has been published in Physical Review B, 111, 085150 – Published 26 February, 2025, we have demonstrated that the hopping Hamiltonian on a Kagome lattice is chiral symmetric, in a sense,
there exists a chiral operator, Γ, that anticommuts with H_Kagome as {H_Kagome, Γ}= H_Kagome Γ- Γ H_Kagome=0. It is a surprise since the nearest neighbor hopping Hamiltonian of the Kagome lattice is not bipartite. The chiral symmetry of the Kagome lattice is hidden! On the other hand, it is natural, since the energy bands of the Kagome lattice is symmetric except the flat band. See also arXiv:2409.04239.