Fig. 2: Wilson loop spectra of 3D magnetic TCIs with (001)-surface MWGs \(p{4}^{{\prime} }{g}^{{\prime} }m\) and \({p}_{c}^{{\prime} }4mm\).

Classification of Wilson loop spectra, whose connectivity is equivalent to the (001)-surface band structure, based on the MCNs \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) and \({{{{\mathcal{C}}}}}_{m}^{y}\). a–e Type-III TMDIs with MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) on the (001) surface. a (001)-surface BZ. \({\widetilde{M}}_{x\overline{y}}\) is the diagonal mirror used to define \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\). b–e Wilson loop spectra corresponding to b \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=0\), c \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=1\), d \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=2\), and e \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=4\). For convenience, the position of the Dirac fermion at \(\overline{M}\) is adjusted to be located at θ(k) = 0. The red (blue) lines correspond to Wilson bands with mirror eigenvalue + i ( − i). The green dashed lines are the reference lines used to count the MCN \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\). In c–e, in which \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\,\ne\, 0\), the dispersion of the Dirac fermion can be chiral in each mirror sector along \(\overline{\Gamma }\)-\(\overline{M}\)-\(\overline{{\Gamma }^{{\prime} }}\). Note that \(\overline{{\Gamma }^{{\prime} }}=(2\pi ,2\pi )\). In b, where \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=0\), the Dirac fermion is nonchiral. In d, where \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=2\), the reference line is crossed by two chiral (upward) modes with mirror eigenvalue + i. In c, where \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=1\), the dispersion is chiral along the entire \(\overline{\Gamma }\)-\(\overline{M}\)-\(\overline{{\Gamma }^{{\prime} }}\) line but locally looks nonchiral near the Dirac point. The Wilson loop structure in c can be deformed into that in h by pushing the twofold crossing at \(\overline{\Gamma }\) upward, which gives a locally chiral dispersion at \(\overline{M}\). When \(| {{{{\mathcal{C}}}}}_{m}^{x\overline{y}}| \,>\, 2\), a Dirac fermion must appear with other gapless surface states along the \(\overline{\Gamma }\)-\(\overline{M}\)-\(\overline{{\Gamma }^{{\prime} }}\) line [black arrows in e]. f–j Type-IV TMDIs with MWG \({p}_{c}^{{\prime} }4mm\) on the (001) surface. f (001)-surface BZ. g–j Wilson loop spectra corresponding to g \(({{{{\mathcal{C}}}}}_{m}^{x\overline{y}},{{{{\mathcal{C}}}}}_{m}^{y})=(0,0)\), h \(({{{{\mathcal{C}}}}}_{m}^{x\overline{y}},{{{{\mathcal{C}}}}}_{m}^{y})=(1,-1)\), i \(({{{{\mathcal{C}}}}}_{m}^{x\overline{y}},{{{{\mathcal{C}}}}}_{m}^{y})=(2,0)\), and j \(({{{{\mathcal{C}}}}}_{m}^{x\overline{y}},{{{{\mathcal{C}}}}}_{m}^{y})=(0,2)\). Note that \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}={{{{\mathcal{C}}}}}_{m}^{y}\) (mod 2) holds for insulators.