Chiral symmetry in non-Hermitian systems: Product rule and Clifford algebra

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