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

Abstract

We show that the spectral weights ${W}_{m\stackrel{P\vec}{K}}(\stackrel{P\vec}{k})$ used for the unfolding of two-component spinor eigenstates $|{\ensuremath{\psi}}_{m\stackrel{P\vec}{K}}^{\mathrm{SC}}\ensuremath{\rangle}=|\ensuremath{\alpha}\ensuremath{\rangle}|{\ensuremath{\psi}}_{m\stackrel{P\vec}{K}}^{\mathrm{SC}},\ensuremath{\alpha}\ensuremath{\rangle}+|\ensuremath{\beta}\ensuremath{\rangle}|{\ensuremath{\psi}}_{m\stackrel{P\vec}{K}}^{\mathrm{SC}},\ensuremath{\beta}\ensuremath{\rangle}$ can be decomposed as the sum of the partial spectral weights ${W}_{m\stackrel{P\vec}{K}}^{\ensuremath{\mu}}(\stackrel{P\vec}{k})$ calculated for each component $\ensuremath{\mu}=\ensuremath{\alpha},\ensuremath{\beta}$ 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 ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\stackrel{P\vec}{K}}(\stackrel{P\vec}{k};\phantom{\rule{0.16em}{0ex}}\ensuremath{\varepsilon})$, which unfolds the primitive cell expectation values ${\ensuremath{\varphi}}^{\mathrm{pc}}(\stackrel{P\vec}{k};\ensuremath{\varepsilon})$ of any arbitrary operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varphi}}$ according to ${\ensuremath{\varphi}}^{\mathrm{pc}}(\stackrel{P\vec}{k};\ensuremath{\varepsilon})=\text{Tr}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\stackrel{P\vec}{K}}(\stackrel{P\vec}{k};\phantom{\rule{0.16em}{0ex}}\ensuremath{\varepsilon})\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varphi}})$. 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.