Chiral symmetry in non-Hermitian systems: Product rule and Clifford algebra
Jose D. H. Rivero, Li Ge
Abstract
Chiral symmetry provides the symmetry protection for a large class of topological edge states. It exists in non-Hermitian systems as well, and the same anticommutation relation between the Hamiltonian and a linear chiral operator, i.e., ${H,\mathrm{\ensuremath{\Pi}}}=0$, now warrants a symmetric spectrum about the origin of the complex energy plane. Utilizing two general approaches to identify and generate chiral symmetry, we first show that its symmetry operator in non-Hermitian systems can go beyond simple spatial transformations such as parity or rotation and include imaginary gauge transformations in a systematic way. Furthermore, we reveal hidden non-Hermitian chiral symmetry and its associated particle-hole symmetry, where their operators take unfamiliar forms due to the presence of energy nonconserving elements. Finally, our implementation of non-Hermitian chiral symmetry in a topological lattice leads to an edge state with ``folded'' localization, where its tail is reflected by the opposite edge and resides on a separate sublattice.