Pancharatnam-Zak Phase for Two-Dimensional Fermionic Systems

Topological states of matter are usually classified by quantized topological invariants. In one-dimensional crystals, the Zak phase is a useful invariant to detect topological states. However, Zak phase is gauge-dependent, and this dependence can sometimes lead to mistakes in the detection of topological phases in more complex systems. To prevent such a mistake, we should look for gauge-invariant topological numbers. In this work we define Pancharatnam-Zak phase, a gauge independent topological number for two-dimensional fermionic systems, and investigate the phase diagram of two topological systems: i) the Su-Schrieffer-Heeger, and ii) the Kitaev models. In two-dimensional systems, the Chern number is usually used to examine the topological phases, however in the presence of time-reversal and inversion symmetries, the Chern number is vanishing, and we have to employ other invariants. We show that the Phancharatnam-Zak phase can be employed for characterizing the topological phases of two-dimensional fermionic systems.