Table 1 The symmetry representation of the 2D and 3D s–d-hybridized models in the main text (Eqs. (1), (3), (4), (5), and (6)).
From: Strong and fragile topological Dirac semimetals with higher-order Fermi arcs
Symmetries of 2D Hamiltonians \( {\mathcal{H}}({k}_{x},{k}_{y}) \) | |
|---|---|
g | \( g{\mathcal{H}}(g{k}_{x},g{k}_{y}){g}^{-1} \) |
Mx | \( {\sigma }^{x}{\mathcal{H}}(-{k}_{x},{k}_{y}){\sigma }^{x} \) |
My | \( {\sigma }^{y}{\mathcal{H}}({k}_{x},-{k}_{y}){\sigma }^{y} \) |
C4z | \( {\tau }^{z}\left(\frac{{{\mathbb{1}}}_{\sigma }-i{\sigma }^{z}}{\sqrt{2}}\right){\mathcal{H}}({k}_{y},-{k}_{x}){\tau }^{z}\left(\frac{{{\mathbb{1}}}_{\sigma }+i{\sigma }^{z}}{\sqrt{2}}\right) \) |
Mz | \( {\sigma }^{z}{\mathcal{H}}({k}_{x},{k}_{y}){\sigma }^{z} \) |
\( {\mathcal{I}} \) | \( {\mathcal{H}}(-{k}_{x},-{k}_{y}) \) |
\( {\mathcal{T}} \) | \( {\sigma }^{y}{{\mathcal{H}}}^{* }(-{k}_{x},-{k}_{y}){\sigma }^{y} \) |
Symmetries of 3D Hamiltonians \( {\mathcal{H}}({k}_{x},{k}_{y},{k}_{z}) \) | |
|---|---|
g | \( g{\mathcal{H}}(g{k}_{x},g{k}_{y},g{k}_{z}){g}^{-1} \) |
Mx | \( {\sigma }^{x}{\mathcal{H}}(-{k}_{x},{k}_{y},{k}_{z}){\sigma }^{x} \) |
My | \( {\sigma }^{y}{\mathcal{H}}({k}_{x},-{k}_{y},{k}_{z}){\sigma }^{y} \) |
C4z | \( {\tau }^{z}\left(\frac{{{\mathbb{1}}}_{\sigma }-i{\sigma }^{z}}{\sqrt{2}}\right){\mathcal{H}}({k}_{y},-{k}_{x},{k}_{z}){\tau }^{z}\left(\frac{{\mathbb{1}}_{\sigma }+i{\sigma }^{z}}{\sqrt{2}}\right) \) |
Mz | \( {\sigma }^{z}{\mathcal{H}}({k}_{x},{k}_{y},-{k}_{z}){\sigma }^{z} \) |
\( {\mathcal{I}} \) | \( {\mathcal{H}}(-{k}_{x},-{k}_{y},-{k}_{z}) \) |
\( {\mathcal{T}} \) | \( {\sigma }^{y}{{\mathcal{H}}}^{* }(-{k}_{x},-{k}_{y},-{k}_{z}){\sigma }^{y} \) |
