Topological properties of a non-Hermitian quasi-one-dimensional chain with a flat band

We investigate the spectral properties of a non-Hermitian quasi-one-dimensional lattice in two possible dimerization configurations. Specifically, we focus on a non-Hermitian diamond chain that presents a zero-energy flat band. The flat band originates from wave interference and results in eigenstates with a finite contribution only on two sites of the unit cell. To achieve the non-Hermitian characteristics, we introduce non-reciprocal intrasite hopping terms in the chain. This leads to the accumulation of eigenstates on the boundary of the system, known as the non-Hermitian skin effect. Despite this accumulation of eigenstates, for one of the two possible configurations, we can characterize the presence of non-trivial edge states at zero energy by a real-space topological invariant known as the biorthogonal polarization. We show that this invariant, evaluated using the destructive interference method, characterizes the non-trivial phase of the non-Hermitian diamond chain. For the other possible non-Hermitian configuration, we find that there is a finite quantum metric associated with the flat band. Additionally, we observe the skin effect despite having the system a purely real or imaginary spectrum. For both configurations, we show that two non- Hermitian diamond chains can be mapped into two models of the Su-Schrieffer-Heeger chains, either non-Hermitian and Hermitian, in the presence of a flat band. This mapping allows us to draw valuable insights into the behavior and properties of these systems.