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                \begin{document}$${\vec {k}} \in {\mathcal {B}}_{L}$$\end{document}$. An explicit computation gives:$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} {{\hat{H}}}( {\vec {k}}) = \begin{pmatrix} -2t_2\alpha _1({\vec {k}})\cos \phi + m({\vec {k}})-\mu &{} -t_{1} \Omega ^*( {\vec {k}}) \\ - t_{1}\Omega ( {\vec {k}}) &{} -2t_2\alpha _1({\vec {k}})\cos \phi - m({\vec {k}})-\mu \end{pmatrix} \end{aligned}$$\end{document}$$where:$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} \begin{aligned}&\alpha _{1}({\vec {k}}) = \sum _{j=1}^3\cos ({\vec {k}}\cdot {\vec {\gamma }}_j)\;,\qquad m({\vec {k}}) = W - 2t_{2}\sin \phi \, \alpha _{2}({\vec {k}})\;, \\&\alpha _{2}({\vec {k}}) = \sum _{j=1}^3\sin ({\vec {k}}\cdot {\vec {\gamma }}_j) \;,\qquad \Omega ({\vec {k}}) = 1 + e^{-i{\vec {k}}\cdot {\vec {\ell }}_1} + e^{-i{\vec {k}}\cdot {\vec {\ell }}_2}\;. \end{aligned} \end{aligned}$$\end{document}$$The corresponding energy bands are$$\documentclass[12pt]{minimal}
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