= Valley masses of the Haldane model
{title2=$m_\tau=M-\tau3\sqrt3t_2\sin\phi$}
The only zeros of $f$, modulo the <reciprocal lattice>, are the two <Dirac valleys in graphene>,
$$
\mathbf K_\tau=\left(\tau\frac{4\pi}{3\sqrt3a},0\right),\qquad \tau=\pm1.
$$
At these momenta $\sum_j\sin(\mathbf K_\tau\cdot\mathbf b_j)=\tau3\sqrt3/2$. Thus the two <Dirac mass> coefficients, measured in <energy> units, are $m_\tau=M-\tau\mu$, with $\mu=3\sqrt3t_2\sin\phi$. For $\mathbf k=\mathbf K_\tau+\mathbf q$,
$$
H_\tau-d_0(\mathbf K_\tau)I=\hbar v(\tau q_x\sigma_x+q_y\sigma_y)+m_\tau\sigma_z+O(|\mathbf q|^2),
\qquad v=\frac{3t_1a}{2\hbar}.
$$
The two <Dirac cones> have opposite orientation, encoded by $\tau$. A scalar sublattice offset $M$ gives equal-sign <Dirac masses>; imaginary second-neighbour hopping at $M=0$ gives opposite-sign <Dirac masses>. Both mechanisms open valley gaps
$$
\Delta_\tau=E_+(\mathbf K_\tau)-E_-(\mathbf K_\tau)=2|m_\tau|.
$$
An observed <band gap> alone therefore cannot establish a <Chern insulator>. In particular, the gapless <nearest-neighbour model of graphene> and its sublattice-mass deformation should not be confused with the topologically nontrivial <Haldane model>.
A <band gap> closes when all three components of $\mathbf d$ vanish. Since $d_x=d_y=0$ only at the two valleys, the phase boundaries are exactly $M=\pm3\sqrt3t_2\sin\phi$. Usually one valley becomes massless; at $M=0$, $\sin\phi=0$, both do. The occupied-band <Chern number> is undefined at every such touching, even if a numerical momentum grid misses its location.
\Image[/haldane-chern-phase-and-valley-gaps.png]
{title=Haldane phases meet where a valley gap closes}
{description=Original model calculation with the stated lower-band Berry-connection convention and $t_2/t_1=0.08$. Left: integer Chern numbers; black curves denote singular parameters, where no Chern number is assigned. Right: the two valley gaps along $\phi=\pi/2$; the vertical axis shows valley gaps, rather than an assumption about global band extrema. No experimental data are plotted.}
{source=haldane-chern-phase-and-valley-gaps.py}
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