= Haldane model
{c}
{title2=$H(\mathbf k)=d_0(\mathbf k)I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$}
{wiki}
<F. Duncan M. Haldane>'s 1988 <tight-binding model> adds complex second-neighbour hopping and opposite sublattice offsets to a <honeycomb lattice>. Directed phases break <time-reversal symmetry> while each primitive cell has zero net <magnetic flux>. Smaller loops can enclose <magnetic flux>; ordinary <Bloch states> remain available.<Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the Parity Anomaly by F. Duncan M. Haldane>[Haldane (1988), pp. 2015–2016, Fig. 1 and Eq. (1)]{ref}
Extend the <nearest-neighbour model of graphene>, ignoring physical <spin> and interactions. With bond length $a$, $t_1>0$, $t_2\geq0$ and ordered sublattice basis $(A,B)$, choose
$$
\begin{aligned}
\boldsymbol\delta_1&=a(0,1),&
\boldsymbol\delta_2&=a(\sqrt3/2,-1/2),&
\boldsymbol\delta_3&=a(-\sqrt3/2,-1/2),\\
\mathbf b_1&=\boldsymbol\delta_3-\boldsymbol\delta_2,&
\mathbf b_2&=\boldsymbol\delta_1-\boldsymbol\delta_3,&
\mathbf b_3&=\boldsymbol\delta_2-\boldsymbol\delta_1,\\
f(\mathbf k)&=\sum_{j=1}^3e^{i\mathbf k\cdot\boldsymbol\delta_j}.&&&&
\end{aligned}
$$
The following <Bloch Hamiltonian> defines $\phi$; reversing directed second-neighbour hopping sends $\phi\mapsto-\phi$:
$$
\begin{aligned}
d_0&=2t_2\cos\phi\sum_j\cos(\mathbf k\cdot\mathbf b_j),\\
d_x&=-t_1\operatorname{Re}f,&d_y&=t_1\operatorname{Im}f,\\
d_z&=M-2t_2\sin\phi\sum_j\sin(\mathbf k\cdot\mathbf b_j),\\
H&=\begin{pmatrix}d_0+d_z&-t_1f\\-t_1f^*&d_0-d_z\end{pmatrix},&
E_\pm&=d_0\pm\sqrt{t_1^2|f|^2+d_z^2}.
\end{aligned}
$$
The <Pauli matrices> act on sublattice amplitudes, as in <sublattice pseudospin in graphene>. The <identity matrix> term $d_0I$ shifts both <energy eigenvalues> equally at each momentum; it changes neither the <eigenvectors> nor the direct separation $E_+-E_-$. A genuine <band insulator> additionally needs $\min_{\mathbf k}E_+>\max_{\mathbf k}E_-$. We use $t_2/t_1=0.08$ in the figures, within the original paper's sufficient nonoverlap restriction $|t_2/t_1|<1/3$.<Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the Parity Anomaly by F. Duncan M. Haldane>[Haldane (1988), p. 2016, immediately after Eq. (1): nonoverlapping bands]{ref}
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