Asymptotic anatomy of the Berry phase for scalar waves in two-dimensional periodic continua

We deploy an asymptotic model for the interaction between nearby dispersion surfaces and respective eigenstates towards explicit evaluation of the Berry phase governed by the scalar wave equation in two-dimensional periodic media. The model, featuring a pair of coupled Dirac equations, entails a four-dimensional parametric space and endows the interacting Bloch eigenstates with an explicit gauge that caters for analytical integration in the wavenumber domain. Among the featured parameters, the one ( s ∈ [ 0 , 1 2 ] ) that synthesizes the phase information on the coupling term is shown to decide whether the Berry connection round the loop is singular ( s = 0 ) or analytic ( s > 0 ). The analysis demonstrates that the Berry phase for two-dimensional lattices is π -quantal and topological when s = 0 , equalling π when the contour encloses a Dirac point and zero in all other situations (avoided crossings or line crossings). The analogous result is obtained, up to an O ( s ) residual, when s ≃ 0 and similarly for s ≃ 1 2 . In the interior of the s -domain, we find that the Berry phase either approximately equals π or is not quantal. Beyond shedding light on the anatomy of the Berry phase in periodic continua, the analysis carries a practical benefit as it permits a single-wavenumber evaluation of this geometrical phase quantity. The asymptotic estimates of the Berry phase are found to be in agreement with their numerical counterparts. For generality, we also include an application to a Dirac-like, three-energy-level system.