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                \begin{document}$$\mathbf{k}'$$\end{document}$ and then setting $\documentclass[12pt]{minimal}
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                \begin{document}$$\mathbf{k}'=\mathbf{0}$$\end{document}$, we get:$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} \partial _{\mathbf{k}'}{{\hat{W}}}_{2;\omega }^{(0)}(\mathbf{0})= & {} e^{-i\frac{2\pi }{3}\omega n_-}T\partial _{\mathbf{k}'}{{\hat{W}}}_{2;\omega }^{(0)}(\mathbf{0})e^{i\frac{2\pi }{3}\omega n_-}=(-R_v)\partial _{\mathbf{k}'} {{\hat{W}}}_{2;\omega }^{(0)*}(\mathbf{0})\nonumber \\= & {} (-P)\partial _{\mathbf{k}'}{{\hat{W}}}_{2;\omega }^{(0)\dagger }(\mathbf{0}), \end{aligned}$$\end{document}$$where $\documentclass[12pt]{minimal}
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                \begin{document}$$R_v$$\end{document}$ (resp. *P*) is the diagonal matrix with diagonal elements $\documen