Experimental scheme for determining the Berry phase in two-dimensional quantum materials with a flat band

Experimentally feasible methods to determine the Berry phase, a fundamental quantity characterizing a quantum material, are often needed in applications. We develop an approach to detecting the Berry phase by using a class of two-dimensional (2D) Dirac materials with a flat band, the $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ lattices. The properties of this class of quantum materials are controlled by a single parameter $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1$, where the left and right end points correspond to graphene with pseudospin-$\frac{1}{2}$ and the dice lattice with pseudospin-1 Dirac-Weyl quasiparticles, respectively, and each specific value of $\ensuremath{\alpha}$ represents a material with a unique Berry phase. Applying a constant electric field to the $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ lattice, we calculate the resulting electric current and find a one-to-one correspondence between the current and the Berry phase in both the linear and nonlinear response regimes. In the linear (Kubo) regime, the main physics is the Zitterbewegung effect. In the nonlinear regime, the Schwinger mechanism dominates. Beyond the nonlinear regime, Bloch-Zener oscillations can arise. Measuring the current thus provides an effective and experimentally feasible way to determine the Berry phase for this spectrum of 2D quantum materials.