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

• Published in: Physical review. B, Condensed matter and materials physics Condensed matter and materials physics, 2015-01, Vol.91 (4), p.041116(R), Article 041116
• Main Authors: Medeiros, Paulo V. C., Tsirkin, Stepan S., Stafström, Sven, Björk, Jonas
• Format: Article
• Language: English
• Subjects: Band structure of solids; Brackets; Condensed matter; Eigenfunctions; Joining; Mathematical analysis; Operators; Spectra; Wave functions
• ISSN: 1098-0121; 1550-235X; 1550-235X
• DOI: 10.1103/PhysRevB.91.041116

We show that the spectral weights W sub(mK) (k) used for the unfolding of two-component spinor eigenstates [psi] super(SC)mK[right angle bracket] = alpha [right angle bracket] [psi] super(SC)mK, alpha [right angle bracket] + beta [right angle bracket] [psi] super(SC)mK, beta [right angle bracket] can be decomposed as the sum of the partial spectral weights W super( mu )mK (k) 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] sub(K) (k; epsilon ), which unfolds the primitive cell expectation values [phi] super(pc) (k; epsilon ) of any arbitrary operator [phi] according to [phi] super(pc) (k; epsilon ) = Tr([rho]K (k; epsilon ) [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.