## 3 Tight-Binding Model  
In this section, we consider the electronic band structure of TMDs, in the whole BZ, from a Slater-Koster tight-binding approximation [49]. We use the model developed by [PERSON] et al. [35], which contains 11 orbitals per layer. In particular, the model contains the five \(d\) orbitals of the metal \(M\) atom and the six \(p\) orbitals of the two chalcogen \(X\) atoms in the unit cell. The used scheme has been recently used in other works studying the electronic band structure of TMDs from a tight-binding perspective [50; 51]. The corresponding base can be expressed as [35]  
\[\left(p_{x}^{t},p_{y}^{t},p_{z}^{t},d_{3z^{2}-r^{2}},d_{xz},d_{yz},\ d_{x^{2}-y^{2}},\ d_{xy},p_{x}^{b},\ p_{y}^{b},\ p_{z}^{b}\right) \tag{1}\]  
where the indices \(t\) and \(b\) label the top and bottom chalcogen planes, respectively. The model is defined by the hopping integrals between the different orbitals, which are described in terms of \(\sigma\), \(\pi\) and \(\delta\) ligands. In the following, we reproduce the most important results, and we refer the reader to Reference [35; 48] for details of the model. The Slater-Koster parameters that account for the relevant hopping processes of the model are \(V_{pd\sigma}\) and \(V_{pd\pi}\) for \(M\)-\(X\) bonds, \(V_{dd\sigma}\), \(V_{dd\pi}\) and \(V_{dd\delta}\) for \(M\)-\(M\) bonds, and \(V_{pp\sigma}\) and \(V_{pp\pi}\) for \(X\)-\(X\) bonds. Additional parameters of the theory are the crystal fields \(\Delta_{0}\), \(\Delta_{1}\), \(\Delta_{2}\), \(\Delta_{p}\), \(\Delta_{z}\), describing, respectively, the atomic level of the \(l=0\) (\(d_{3z^{2}-r^{2}}\)), the \(l=1\) (\(d_{xz}\), \(d_{yz}\)), the \(l=2\) (\(d_{x^{2}-y^{2}}\), \(d_{xy}\)) \(M\) orbitals, the in-plane (\(p_{x}\), \(p_{y}\)) \(X\) orbitals and the out-of-plane \(p_{z}\)\(X\) orbitals. This model can be simplified by performing a unitary transformation that takes the \(p\) orbitals of the top and bottom layers of the \(X\) atoms into their symmetric and antisymmetric combinations with respect to the \(z\)-axis. This way, the 11-band model is decoupled into a \(6\times 6\) block with even (odd)  
Figure 4: Band structure and orbital character of single-layer MoS\({}_{2}\). The thickness of the bands represents the orbital weight, where the \(d\) character (\(d_{2}=d_{x^{2}-y^{2}},d_{xy}\) and \(d_{0}=d_{3z^{2}-r^{2}}\)) refers to the Mo atom \(4d\) orbitals, while the \(p\) character (where \(p_{xy}=p_{x},p_{y}\)) refers to \(3p\) orbitals of sulfur. Top panels correspond to orbital weight from DFT calculations, whereas bottom panels correspond to orbital weight from TB results. The **black** lines in the bottom panels are the DFT bands. Notice that spin–orbit coupling is not included in these plots. symmetry of the \(p_{x},p_{y}\) (\(p_{z}\)) orbitals with respect to \(z\rightarrow-z\) inversion, and a \(5\times 5\) band block with opposite combination. Since low energy excitations belong exclusively to the first block, the fit to DFT that we will present later will be performed within this sector. Therefore, the relevant bands above and below the gap are well accounted for by the reduced Hilbert space:  
\[\psi=\left(d_{3z^{2}-r^{2}},\;d_{x^{2}-y^{2}},\;d_{xy},p_{x}^{S},\;p_{y}^{S},\;p _{z}^{A}\right), \tag{2}\]  
where the \(S\) and \(A\) superscripts stand for the symmetric and antisymmetric combinations of the \(p\)-orbitals of the \(X\) atom, \(p_{i}^{S}=1/\sqrt{2}(p_{i}^{i}+p_{i}^{b})\) and \(p_{i}^{A}=1/\sqrt{2}(p_{i}^{i}-p_{i}^{b})\), with \(i=x,y,z\). The tight-binding Hamiltonian defined by base (2), including local spin-orbit coupling, can be expressed in real space as  
\[H = \sum_{i,\mu\
u}\epsilon_{i,\mu\
u}c_{i,\mu}^{\dagger}c_{i,\
u}+ \sum_{ij,\mu\
u}[t_{ij,\mu\
u}c_{i,\mu}^{\dagger}c_{j,\
u}+\text{H.c.}], \tag{3}\]  
where \(c_{i,\mu}^{\dagger}(c_{i,\mu})\) creates (annihilates) an electron in the unit cell \(i\) in the atomic orbital \(\mu=1,\ldots,6\), belonging to the Hilbert space defined by base (2).