Semiclassics for matrix Hamiltonians: The Gutzwiller trace formula and applications to the graphene-type systems

Abstract: We have extended the semi-classical theory to include a general account of matrix valued Hamiltonians, i.e. those that describe quantum systems with internal degrees of freedoms, based on a generalization of the Gutzwiller trace formula for a $n\times n$ dimensional Hamiltonian $H(\hat{\boldsymbol p},\hat{\boldsymbol q})$. The classical dynamics is governed by $n$ Hamilton-Jacobi (HJ) equations, that act in a phase space endowed with a classical Berry curvature encoding anholonomy in the parallel transport of the eigenvectors of $H(\hat{\boldsymbol p}\to\boldsymbol p,\hat{\boldsymbol q}\to\boldsymbol q)$, which describe the internal structure of the semi-classical particles. This Berry curvature is a fully classical object and is, in that sense, as fundamental to the semi-classical theory of matrix Hamiltonians as the Hamilton-Jacobi equations. At the $\mathcal{O}(\hbar^1)$ level, it results in an additional semi-classical phase composed of (i) a Berry phase and (ii) a dynamical phase resulting from the classical particles "moving through the Berry curvature". We show that the dynamical part of this semi-classical phase will, generally, only be zero only for the case in which the Berry phase is topological (i.e. depends only on the winding number). We illustrate the method by calculating the Landau spectrum for monolayer graphene, the four-band model of AB bilayer graphene, and for a more complicated matrix Hamiltonian describing the silicene band structure. Finally we apply our method to an inhomogeneous system consisting of a strain engineered one dimensional moir\'e in bilayer graphene, finding localized states near the Dirac point that arise from electron trapping in a semi-classical moir\'e potential. The semi-classical density of states of these localized states we show to be in perfect agreement with an exact quantum mechanical calculation of the density of states.