Matrix representation of trivial Hamiltonian

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 $1/2$.