Gauge freedom for degenerate Bloch states

In nonmagnetic crystals with inversion symmetry the electronic bands are twofold degenerate. As a consequence, any orthonormalized linear combination of the two corresponding eigenfunctions can represent the electron wave function. A priori it is not obvious which superposition, gauge, should be chosen to calculate a quantity which is not gauge invariant within a certain approximation. Here we consider gauge options appropriate under particular physical conditions.

Figure 1

Left: $z$ component of the diagonal elements of the $\mathfrak{su}\left(2\right)$ spin polarization (solid line) of the lower Kramers doublet within gauge I. Right: $z$ component of the first diagonal element of the corresponding $\mathfrak{su}\left(2\right)$ Berry curvature (solid line) jumps in gauge I.

Figure 2

Left: $z$ component of the diagonal elements of the $\mathfrak{su}\left(2\right)$ spin polarization (solid line) of the lower Kramers doublet in gauge II. Right: $z$ component of the first diagonal element of the corresponding $\mathfrak{su}\left(2\right)$ Berry curvature (solid line) remains continuous within gauge II.