Characterizing the Topological Properties of 1D Non-Hermitian Systems without the Berry-Zak Phase

A new method is proposed to predict the topological properties of 1D periodic structures in wave physics, including quantum mechanics. From Bloch waves, a unique complex valued function is constructed, exhibiting poles and zeros. The sequence of poles and zeros of this function is a topological invariant that can be linked to the Berry-Zak phase. Since the characterization of the topological properties is done in the complex plane, it can easily be extended to the case of non-Hermitian systems. The sequence of poles and zeros allows to predict topological phase transitions. Topological properties of photonic crystals or insulators are generally addressed by means of integer numbers obtained, for example, through the Berry connection. A completely different approach is proposed here : a 1D structure can be characterized by means of the poles and zeros of a function. The approach applies to non-Hermitian as well as disordered structures.image