# Definition The sewing Matrix for [[Topological Insulator|topological insulators]] (TIs) is defined as $ w_{mn} = \braket{u_{m,-\pmb{k}}|\mathcal{S}|u_{n,\pmb{k}}} $ where $\mathcal{S}$ is the symmetry operator of the TI and $\ket{\psi_n(\pmb{k})}$ is the [[Bloch's Theorem|Bloch eigenstate]] of [[Band Structure|band]] $n$ with [[crystal momentum]] $\pmb{k}$. For a [[Time-Reversal Symmetric Topological Insulator| time-reversal symmetric topological insulator (TRS-TI)]] $\mathcal{S} = \Theta$ where $\Theta$ is the time-reversal operator. For an [[Antiferromagnetic Topological Insulator|antiferromagnetic topological insulator (AFTI)]], $\mathcal{S} = \Theta T_{\pmb{D}}$ where $T_{\pmb{D}}$ is the translation operator by a vector $\pmb{D}$. A similar formula applies for special classes of [[Higher-Order Topological Insulator|higher-order topological insulators (HOTI)]]. Using the sewing matrix, we can compute $\ket{u_{n,-\pmb{k}}}$ as: $ \begin{align} \sum_n w^*_{mn}(\pmb{k})\Theta \ket{u_{n,\pmb{k}}} = \ket{u_{m,-\pmb{k}}} ,\end{align} $ # Properties $w(\pmb{k})$ has the following property: $ \begin{align} w_{mn}(\pmb{k}) &= - w_{nm}(-\pmb{k}) .\end{align} $ It is also easy to show that $w(\pmb{k})$ is a unitary matrix: $ \begin{align} \left(w^{\dagger}(\pmb{k}) w(\pmb{k})\right)_{mn} &= \sum_{l} w^{\dagger}_{ml}(\pmb{k}) w_{ln}(\pmb{k}) \\ &=\sum_{l} \bra{u_{l,\pmb{k}}}\Theta^{\dagger}\ket{u_{m,-\pmb{k}}}\times\bra{u_{l,-\pmb{k}}}\Theta\ket{u_{n,\pmb{k}}},\\ &= \delta_{mn} .\end{align} $ Thus $w^\dagger(\pmb{k}) w(\pmb{k}) = I$ and $w(\pmb{k})$ is unitary.