I was reading Kitaev 2009 periodic table paper when I came across the following
"Let us define the trivial hamiltonian:" $$ \hat{H}_{\text {triv }}=\sum_{j}\left(\hat{a}_{j}^{\dagger} \hat{a}_{j}-\frac{1}{2}\right)=\hat{H}_{Q} $$ where $$ Q=\left(\begin{array}{ccccc} 0 & 1 & & & \\ -1 & 0 & & & \\ & & 0 & 1 & \\ & & -1 & 0 & \\ & & & & \ddots \end{array}\right) $$ Now I wonder what this $Q$ matrix is all about, since I think the Hamiltonian matrix should be a diagonal matrix with diagonal value being $1/2$.