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. Specifically, at high-symmetry points, the dispersion relation between energy and momentum can often be approximated as linear (or nearly linear), leading to an effective Dirac or Weyl Hamiltonian, where the Berry curvature is typically computed without accounting for the breakdown of the adiabatic condition. In this work, we demonstrate that time-dependent topological quantities-specifically the Aharonov–Anandan phase-can be employed to extract information not only about topological properties but also about band transitions across a wide range of systems, including gapless 2D materials, topological insulators, Weyl semimetals, and tilted Dirac materials. In particular, we establish a relationship between the current and the Aharonov–Anandan phase, demonstrating that photon-induced transitions give rise to current vortices. To illustrate this, we analyze graphene under electromagnetic radiation from a time-dependent perspective, showing how the Aharonov–Anandan and Berry phases provide complementary insights into topology, interband transitions, and currents. This analysis is carried out using the Dirac–Bloch formalism and by solving the time-dependent equations within the framework of Floquet theory.