Unfolding spinor wave functions and expectation values of general operators: Introducing the unfolding-density operator

We show that the spectral weights W-m (K) over right arrow((k) over right arrow) used for the unfolding of two-component spinor eigenstates vertical bar psi(SC)(m (K) over right arrow)> = vertical bar alpha > vertical bar psi(SC,alpha)(m (K) over right arrow) > + vertical bar beta > vertical bar psi(SC,beta)(m (K) over right arrow)> can be decomposed as the sum of the partial spectral weights W-m (K) over right arrow(mu)((k) over right arrow) calculated for each component mu = alpha, beta independently, effortlessly turning a possibly complicated problem involving two coupled quantities into two independent problems of easy solution. Furthermore, we define the unfolding-density operator (rho) over cap ((K) over right arrow)((k) over right arrow; epsilon), which unfolds the primitive cell expectation values phi(pc)((k) over right arrow; epsilon) of any arbitrary operator (phi) over cap according to phi(pc)((k) over right arrow; epsilon) = Tr((rho) over cap ((K) over right arrow)((k) over right arrow; epsilon) (phi) over cap). As a proof of concept, we apply the method to obtain the unfolded band structures, as well as the expectation values of the Pauli spin matrices, for prototypical physical systems described by two-component spinor eigenfunctions.