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

$${\mathit{\delta }}_1=\frac{a}{2}(1,\sqrt{3}),{\mathit{\delta }}_2=\frac{a}{2}(1,-\sqrt{3}),{\mathit{\delta }}_3=-a(1,0),\backslash boldsymbol\backslash delta\_1=\backslash frac\{a\}\{2\}(1,\backslash sqrt\{3\}),\backslash qquad\; \backslash boldsymbol\backslash delta\_2=\backslash frac\{a\}\{2\}(1,-\backslash sqrt\{3\}),\backslash qquad\; \backslash boldsymbol\backslash delta\_3=-a(1,0),$$

and

$${\mathbf{g}}_1={\mathbf{a}}_1-{\mathbf{a}}_2,{\mathbf{g}}_2=-{\mathbf{a}}_1,{\mathbf{g}}_3={\mathbf{a}}_2,\backslash mathbf\; g\_1=\backslash mathbf\; a\_1-\backslash mathbf\; a\_2,\backslash qquad\; \backslash mathbf\; g\_2=-\backslash mathbf\; a\_1,\backslash qquad\; \backslash mathbf\; g\_3=\backslash mathbf\; a\_2,$$

where

$${\mathbf{a}}_1=\frac{a}{2}(3,\sqrt{3}),{\mathbf{a}}_2=\frac{a}{2}(3,-\sqrt{3})\mathrm{.}\backslash mathbf\; a\_1=\backslash frac\{a\}\{2\}(3,\backslash sqrt\{3\}),\backslash qquad\; \backslash mathbf\; a\_2=\backslash frac\{a\}\{2\}(3,-\backslash sqrt\{3\}).$$

Use the Bloch Hamiltonian

$$H(\mathbf{k})=\mathbf{d}(\mathbf{k})\cdot \mathit{\sigma },H(\backslash mathbf\; k)=\backslash mathbf\; d(\backslash mathbf\; k)\backslash cdot\backslash boldsymbol\backslash sigma,$$

with

The shared parameters are

$$a=2.683,M=0.026,t_1=-\mathrm{0.1.}a=2.683,\backslash qquad\; M=0.026,\backslash qquad\; t\_1=-0.1.$$

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

$$\mathbf{A}(t)=A_0{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\omega t}{2n_{\mathrm{c}\mathrm{y}\mathrm{c}}}\right)\sin (\omega t){\widehat{\mathbf{e}}}_{\parallel },0\le t\le \frac{2\pi n_{\mathrm{c}\mathrm{y}\mathrm{c}}}{\omega },\backslash mathbf\; A(t)=A\_0\backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash omega\; t\}\{2n\_\{\backslash mathrm\{cyc\}\}\}\backslash right)\; \backslash sin(\backslash omega\; t)\backslash ,\backslash hat\{\backslash mathbf\; e\}\_\{\backslash parallel\},\; \backslash qquad\; 0\backslash le\; t\backslash le\backslash frac\{2\backslash pi\; n\_\{\backslash mathrm\{cyc\}\}\}\{\backslash omega\},$$

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 for the common submission entry points.