Figure 6

Transition from Class α to Class β by breaking \({ {\mathcal M} }_{z}\) mirror symmetry. (a) Crystal structure of WC with the W and C atoms represented by silver and bronze spheres and the k _z  = 0 (\({ {\mathcal M} }_{z}\)) mirror plane by a green plane. (b) Band structure along k _z and around the triply degenerate band crossing for Class α. The blue and black bands denote the doubly and singly degenerate bands, respectively. (c) At the location marked by the yellow dot in (b), we present the band dispersion of the doubly degenerate band along the in-plane k _y -direction. A quadratic dispersion for the \({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\) bands and a touching point between them is observed. The two bands are colored blue to signify they are from the doubly-degenerate band in (b). (d) A trivial Berry phase of 2π was computed for a closed loop around the touching point in (c). (e) The displacement of the W atom in (a) along the c lattice constant direction breaks the \({ {\mathcal M} }_{z}\) mirror symmetry. (f) Similar to (b), but for the broken \({ {\mathcal M} }_{z}\) mirror symmetry case. (f) Along a generic k _y direction, the doubly degenerate band is shown to have transitioned from Class α to Class β occurs though breaking the \({ {\mathcal M} }_{z}\) mirror symmetry. Specifically, the observed linearly dispersing touching points between \({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\) signifies the expected character of Class β. The red circle denotes one of the two linear touching points. (h) Illustration showcasing a non-trivial π Berry phase for a closed loop around the linearly dispersing touching points.