cases/case38/task.md

# Valley-resolved and Brillouin-zone-integrated topological high-harmonic emission

## Physical setting

Work in atomic units. Consider a two-band Haldane model on a hexagonal lattice
with two sites per unit cell. Let

$$
\boldsymbol\delta_1=\frac{a}{2}(1,\sqrt{3}),\qquad
\boldsymbol\delta_2=\frac{a}{2}(1,-\sqrt{3}),\qquad
\boldsymbol\delta_3=-a(1,0),
$$

and

$$
\mathbf g_1=\mathbf a_1-\mathbf a_2,\qquad
\mathbf g_2=-\mathbf a_1,\qquad
\mathbf g_3=\mathbf a_2,
$$

where

$$
\mathbf a_1=\frac{a}{2}(3,\sqrt{3}),\qquad
\mathbf a_2=\frac{a}{2}(3,-\sqrt{3}).
$$

Use the Bloch Hamiltonian

$$
H(\mathbf k)=\mathbf d(\mathbf k)\cdot\boldsymbol\sigma,
$$

with

$$
\begin{aligned}
d_x(\mathbf k)&=t_1\sum_{n=1}^{3}\cos(\mathbf k\cdot\boldsymbol\delta_n),\\
d_y(\mathbf k)&=-t_1\sum_{n=1}^{3}\sin(\mathbf k\cdot\boldsymbol\delta_n),\\
d_z(\mathbf k)&=M+2\,\operatorname{Im}(t_2)
\sum_{n=1}^{3}\sin(\mathbf k\cdot\mathbf g_n).
\end{aligned}
$$

The shared parameters are

$$
a=2.683,\qquad M=0.026,\qquad t_1=-0.1.
$$

Compare two physical cases, $t_2=-0.0013i$ and $t_2=-0.0087i$. Drive the
system along the $\Gamma$-$M$ direction through the Peierls shift
$\mathbf k\mapsto\mathbf k+\mathbf A(t)$, with

$$
\mathbf A(t)=A_0\sin^2\!\left(\frac{\omega t}{2n_{\mathrm{cyc}}}\right)
\sin(\omega t)\,\hat{\mathbf e}_{\parallel},
\qquad 0\le t\le\frac{2\pi n_{\mathrm{cyc}}}{\omega},
$$

and $\mathbf A(t)=0$ outside the pulse. Use
$A_0=0.1$, $\omega=0.0075$, and $n_{\mathrm{cyc}}=5$.

Treat the driven crystal within independent-electron dynamics of this
two-band Bloch Hamiltonian, starting from its lower band. Characterize the
emitted high-harmonic field through its complex components parallel and
perpendicular to the driver. Use one consistent Fourier convention and a
common normalization; absolute spectral intensity is not a grading target.

## Research problem

Determine whether and how the change between the two Haldane-model cases affects the
polarization and helicity of high-harmonic emission, and establish the relation
between the valley-resolved response at the inequivalent $K$ and $K'$ points
and the observable response of a completely occupied lower band.

Use your computed harmonic yields and complex channel phases to decide whether
the valley response follows one common rule at both valleys. Determine whether
any polarization information remains in the integrated
response, and state what that observable can and cannot establish about its
microscopic crystal-momentum origin. Limit all conclusions to the bulk,
independent-electron, two-band model specified above.

Your evidence must keep the two model cases, the $K$ and $K'$ initial states,
the fixed-valley and filled-band calculations, and the two response directions
unambiguous. The derivation, numerical representation, Brillouin-zone sampling,
propagator, time step, differentiation and Fourier procedures, treatment of
weak components, reliability strategy, normalization, plotting choices, and
report organization are for you to decide and justify. No particular number,
order, or layout of figures is prescribed.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing the task-specific derivation, code, numerical
data underlying all material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run
and environment information. No fixed file tree, filename scheme, panel count,
or complete exploration history is required.

Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary are binding. No separate aggregate CPU-hour limit applies. Greater computational expense
receives no additional credit.

## Reference and integrity policy

Complete this task without network access or live literature
retrieval. You may rely on your existing general scientific knowledge and, if any are
explicitly supplied with this task, on those offline materials or
software documentation. Record every such source actually consulted. For formulas,
methods, constants, or conventions used from general knowledge rather than a
consulted source, state enough of the method basis, definition, value, units,
and convention to make the calculation auditable; do not invent a citation.
You are not required to identify or prove a publication date or to retrieve
literature while completing this task.

The hidden target article and its identifying metadata, figures or captions,
supplementary material, author code or data, target-specific reproduction
code, solution or scoring material, online solutions, and worked
reproductions of this case must not be used. All submitted scientific figures
and numerical data must be generated by your own implementation of the stated
problem.

See [submission requirements](../../submission_requirements.md) for the common submission entry points.