I am a math student who is learning topological phases from this website.
Let's assume the fermi level is zero. For the graphene, the sublattice symmetry $\sigma_z H \sigma_z = -H$ makes the Hamiltonian look like
\begin{align*} H = \begin{bmatrix} 0 & H_{AB}\\ H_{AB}^\dagger & 0 \end{bmatrix} \end{align*}
I can see that the spectrum of $H$ is symmetric with respect to the fermi level, but I don't see why it is away from $0$. In my understanding, a Hamiltonian is a hermitian operator that may differ from questions to question. Therefore, for example, the trivial case $H_{AB} = 0$ clearly has $0$ in the spectrum; or when $H_{AB}$ has eigenvalue $0$, then so does $H$.
When physics people write down a Hamiltonian, is there any assumption or convention made without explicitly stated? For instance, $H_{AB}$ is assumed to be positive definite? Or the fact that the spectrum is away from $0$ is based on observation?