CritPt / Challenge_37_main / Consider a single-particle Hamiltonian on a lattice.

# Problem setup:
Consider a single-particle Hamiltonian on a lattice.

Given a generic isolated set of N bands with projector P_(k) constructed by the perodic part of the Bloch states,
we can define the quantum metric as
\begin{equation}
g_{ij}(\boldsymbol{k}) = \frac{1}{2}\mathrm{Tr}[\partial_{k_i} P_{\boldsymbol{k}} \partial_{k_j} P_{\boldsymbol{k}}]\ .
\end{equation}

The gauge invariant part of the Wannier spread of the isolated set of bands is proportional to
\begin{equation}
\mathop{\mathrm{Tr}}\mathcal{G} = \int d^2 k\ \mathop{\mathrm{Tr}}[g(\boldsymbol{k})]\ ,
\end{equation}
where the integration ranges over the first Brillouin zone.

Given the following 2D Hamiltonian,
\begin{align}\begin{split}
H & = \int d^2 r\ c^\dagger_{\boldsymbol{r}} \left[ - \frac{1}{2 m} \nabla^2 + \lambda (- \mathrm{i} \partial_y \sigma_x +  \mathrm{i} \partial_x \sigma_y)  + \Delta_1 \sum_{i=1}^{3} \sum_{s=\pm} e^{ s  \mathrm{i} \boldsymbol{g}_i^{(1)}\cdot\boldsymbol{r}}  \right] c_{\boldsymbol{r}}  \\
& \qquad + \int d^2 r\ c^\dagger_{\boldsymbol{r}} \left[  \mathrm{i} \Delta_2 \sum_{i=1}^{3} \sum_{s=\pm} s e^{ s  \mathrm{i} \boldsymbol{g}_i^{(1)}\cdot\boldsymbol{r}}   +  \sum_{i=1}^{3} \sum_{s=\pm} (\Delta_3 + s  \mathrm{i} \Delta_4)e^{s  \mathrm{i} \boldsymbol{g}_i^{(2)}\cdot\boldsymbol{r}}  \right] c_{\boldsymbol{r}} \ ,
\end{split}\end{align}
where c_(r)^(†)=(c_(r,↑)^(†),c_(r,↓)^(†)), c_(r,s)^(†) creates an electron at 2D coordinate r and with spin s, the primitive reciprocal lattice vectors reads b_(M,1)=(0,1) and b_(M,2)=C₆b_(M,1), and g_(i)^((1))=C₃^(i−1)b_(M,1), and g_(i)^((2))=C₃^(i−1)(b_(M,1)+b_(M,2)).

We choose
\begin{equation}
2 m = 1,\ \lambda = 1.9,\ \Delta_1 = 0.12,\ \Delta_2 = 0.005,\ \Delta_3 = 0.05,\ \Delta_4 = 0.01\ .
\end{equation}


# Main problem:

Is the set of the lowest two bands isolated?
What is the direct energy gap between the lowest two bands?
What is $\frac{1}{2\pi}\mathop{\mathrm{Tr}}\mathcal{G}$ for the set of the lowest two bands?
Is the Kane-Mele time-reversal Z₂ topology of the set of the lowest two bands trivial or nontrivial?
Can the set of the lowest two bands be expressed in terms of two exponentially localized Wannier functions?

For all numerical calculations, please use the 43 shortest reciprocal lattice vectors, use a 60×60 equal-spacing hexagonal-symmetric momentum mesh in the Brillouin zone, and keep four decimal places for all numerical answers.
If any quantity is ill-defined, use N/A as the answer.

# Problem setup:
Consider a single-particle Hamiltonian on a lattice.

Given a generic isolated set of $N$ bands with projector $P_{\boldsymbol{k}}$ constructed by the perodic part of the Bloch states,
we can define the quantum metric as
\begin{equation}
g_{ij}(\boldsymbol{k}) = \frac{1}{2}\mathrm{Tr}[\partial_{k_i} P_{\boldsymbol{k}} \partial_{k_j} P_{\boldsymbol{k}}]\ .
\end{equation}

The gauge invariant part of the Wannier spread of the isolated set of bands is proportional to
\begin{equation}
\mathop{\mathrm{Tr}}\mathcal{G} = \int d^2 k\ \mathop{\mathrm{Tr}}[g(\boldsymbol{k})]\ ,
\end{equation}
where the integration ranges over the first Brillouin zone.

Given the following 2D Hamiltonian,
\begin{align}\begin{split}
H & = \int d^2 r\ c^\dagger_{\boldsymbol{r}} \left[ - \frac{1}{2 m} \nabla^2 + \lambda (- \mathrm{i} \partial_y \sigma_x +  \mathrm{i} \partial_x \sigma_y)  + \Delta_1 \sum_{i=1}^{3} \sum_{s=\pm} e^{ s  \mathrm{i} \boldsymbol{g}_i^{(1)}\cdot\boldsymbol{r}}  \right] c_{\boldsymbol{r}}  \\
& \qquad + \int d^2 r\ c^\dagger_{\boldsymbol{r}} \left[  \mathrm{i} \Delta_2 \sum_{i=1}^{3} \sum_{s=\pm} s e^{ s  \mathrm{i} \boldsymbol{g}_i^{(1)}\cdot\boldsymbol{r}}   +  \sum_{i=1}^{3} \sum_{s=\pm} (\Delta_3 + s  \mathrm{i} \Delta_4)e^{s  \mathrm{i} \boldsymbol{g}_i^{(2)}\cdot\boldsymbol{r}}  \right] c_{\boldsymbol{r}} \ ,
\end{split}\end{align}
where $c^\dagger_{\boldsymbol{r}} = (c^\dagger_{\boldsymbol{r},\uparrow}, c^\dagger_{\boldsymbol{r},\downarrow})$, $c^\dagger_{\boldsymbol{r},s}$ creates an electron at 2D coordinate $\boldsymbol{r}$ and with spin $s$, the primitive reciprocal lattice vectors reads $\boldsymbol{b}_{M,1} = (0,1)$ and $\boldsymbol{b}_{M,2} = C_6 \boldsymbol{b}_{M,1}$, and $\boldsymbol{g}_i^{(1)} = C_3^{i-1}\boldsymbol{b}_{M,1}  $, and $\boldsymbol{g}_i^{(2)} = C_3^{i-1}(\boldsymbol{b}_{M,1} + \boldsymbol{b}_{M,2})  $.

We choose
\begin{equation}
2 m = 1,\ \lambda = 1.9,\ \Delta_1 = 0.12,\ \Delta_2 = 0.005,\ \Delta_3 = 0.05,\ \Delta_4 = 0.01\ .
\end{equation}


# Main problem:

Is the set of the lowest two bands isolated?
What is the direct energy gap between the lowest two bands?
What is $\frac{1}{2\pi}\mathop{\mathrm{Tr}}\mathcal{G}$ for the set of the lowest two bands?
Is the Kane-Mele time-reversal $Z_2$ topology of the set of the lowest two bands trivial or nontrivial?
Can the set of the lowest two bands be expressed in terms of two exponentially localized Wannier functions?

For all numerical calculations, please use the 43 shortest reciprocal lattice vectors, use a $60\times 60$ equal-spacing hexagonal-symmetric momentum mesh in the Brillouin zone, and keep four decimal places for all numerical answers.
If any quantity is ill-defined, use N/A as the answer.