When Adiabaticity Is Not Enough to Study Topological Phases in Solid-State Physics: Comparing the Berry and Aharonov-Anandan Phases in 2D Materials

Abstract

Topological phases emerge as the parameters of a quantum system vary with time. Under the adiabatic approximation, the time dependence can be eliminated, allowing the Berry topological phase to be obtained from a closed trajectory in parameter space. In solid-state physics, this approach is commonly applied by taking a reciprocal space wavevector as the parameter, which is assumed to be varied by electromagnetic fields.The Berry curvature is then obtained by computing the derivatives of Bloch wavefunctions in reciprocal space. However, in many systems-especially gapless ones-the adiabatic approximation is never satisfied. This is particularly true in Dirac and Weyl materials, where the Berry curvature is often calculated without considering the breakdown of the adiabatic condition. In this work, we demonstrate how other time-dependent topological quantities, specifically the Aharonov-Anandan phase, can be used to extract information not only about topology but also about band transitions in 2D materials. In particular, a relationship between the current and the Aharonov-Anandan phase is proved, showing that photon-induced transitions produce current vortices. To illustrate this, we analyze graphene under electromagnetic radiation from a time-driven perspective, showing how the Aharonov-Anandan and Berry phases provide complementary insights into topology, interband transitions, and currents. This is achieved by using the Dirac-Bloch formalism and by solving the time-dependent equations within Floquet theory.