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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 $1/2$.

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    $\begingroup$ He's rewriting the Hamiltonian in terms of Majorana fermions. The Majorana fermion Hamiltonian can be written as a matrix equation $\vec{c} Q \vec{c}^T$ where $\vec c$ is a vector of the Majorana fermions and $Q$ is the matrix you've written above. If this is still unclear, I can write a longer response. $\endgroup$
    – 4xion
    Commented Sep 1, 2020 at 14:57

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It depends on the notation defined in this specific paper: what is $Q$ and what $H_Q$ means?

Regardless, since $Q$ is not the same as $H_Q$, it doesn't have to be diagonal simultaneously with the Hamiltonian. And while a Hamiltonian is diagonal in the representation of its eigenstates, it is certainly not the case in an arbitrary representation.

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