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

We show that the spectral weights $W_{m\vec{K}}\left(\vec{k}\right)$ used for the unfolding of two-component spinor eigenstates $|{\psi }_{m\vec{K}}^{\mathrm{SC}}\rangle =|\alpha \rangle |{\psi }_{m\vec{K}}^{\mathrm{SC}},\alpha \rangle +|\beta \rangle |{\psi }_{m\vec{K}}^{\mathrm{SC}},\beta \rangle$ can be decomposed as the sum of the partial spectral weights $W_{m\vec{K}}^{\mu }\left(\vec{k}\right)$ 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 ${\widehat{\rho }}_{\vec{K}}(\vec{k};\varepsilon )$, which unfolds the primitive cell expectation values ${\phi }^{\mathrm{pc}}(\vec{k};\varepsilon )$ of any arbitrary operator $\widehat{\phi }$ according to ${\phi }^{\mathrm{pc}}(\vec{k};\varepsilon )=\text{Tr}({\widehat{\rho }}_{\vec{K}}(\vec{k};\varepsilon )\widehat{\phi })$. 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.

Figure 1

Graphene@Au: (a) Band structure and projections of the Pauli vector $\vec{\sigma }\equiv {\widehat{\sigma }}_x{\widehat{e}}_x+{\widehat{\sigma }}_y{\widehat{e}}_y+{\widehat{\sigma }}_z{\widehat{e}}_z$ for the $2\times 2$ SC before (a) and after (b) unfolding onto the PC. The inset shows the $2\times 2$ SC used (black rhombus), and a PC (red rhombus). In the curves, blue and red indicate opposite signs for the values of $\vec{\sigma }$ projected perpendicular to the PCBZ wave vectors, and black means a zero net value.

Figure 2

Graphene@Bi: EBS calculated along high-symmetry directions of graphene's PCBZ. The upper inset shows a top view of the atomic structure of the system, with the SC indicated by a black rhombus. The lower inset details the region delimited by the green rectangle. In the EBS inset, the colors are defined as in Fig. 1 and the area of the spheres represents the magnitude of the projections.

Figure 3

Au(110), $2\times 1$ reconstructed surface: EBS along high-symmetry directions of the PCBZ of the nonreconstructed surface. The inset shows a zoom-in into the region delimited by the green rectangle. In the inset, the colors are defined as in Fig. 1, but the projections are now onto the perpendicular to $\vec{k}\text{−}\overline{Y}$. The area of the spheres represents the magnitude of the projections.