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Matrix representation of trivial Hamiltonian

Physics Asked by Jiahao Fan on October 8, 2020

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 $1/2$.

One Answer

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.

Answered by Vadim on October 8, 2020

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