omitted because it simply leads to a redefinition of the zero-energy. Here $\tilde{v}=l^2N_+N_- v$ is the renormalized Fermi velocity, and $l^2$ is the area of $UC$. The effective mass $\mu$ is given by $$\mu=\tfrac{1}{2}\int_{UC}dr^2~ U(\bm{r})\left(N_+^2e^{2F(\bm{r})}- N_-^2e^{-2F(\bm{r})}\right),\label{eq_mass}$$ The signs of $U(\bm{r})$ and $F(\bm{r})$ are said to be correlated when there is a large overlap between regions of positive $U$ and positive $F$ and anti-correlated when there is a large overlap between regions of positive $U$ and negative $F$. When the signs of $U$ and $F$ are either correlated or anti-correlated, the two terms in the integrand tend not to cancel and $\mu$ is non-zero.

The leading order perturbative expressions for the Bloch state $u_{\pm,\bm{k}}(\bm{r})$ in the vicinity of $\bm{k}=0$ is $$u_{\pm,\bm{k}}(\bm{r})=\tfrac{1}{\sqrt{2}}\left[\sqrt{1\pm\tfrac{\mu}{|\varepsilon_{\pm,\bm{k}}|}}\psi_+(\bm{r})
+\sqrt{1\mp\tfrac{\mu}{|\varepsilon_{\pm,\bm{k}|}}}e^{i\theta}\psi_-(\bm{r})\right],
\label{eq_u}$$ with $\theta=\arg(k_x+ik_y)$ and $\varepsilon_{\pm,\bm{k}}=\pm\sqrt{(\hbar v|\bm{k}|)^2+\mu^2}$ the energy of the state.

It is worth emphasizing here that the same effective Hamiltonian $h(\bm{k})$, and therefore the same mass term is induced in both valleys. This is to be contrasted with the mass term that arises when a staggered on-site potential is applied to otherwise clean graphene, by for instance placing the sample on an appropriate substrate.[@Zhou07] If the staggered potential takes on the value $m$ on the $A$ sublattice and 