cases/case05/task.md

Topological edge states in finite-chain high-harmonic generation

Background

In a finite dimerized chain, opposite structural phases can have different edge-state content even when both are insulating. High-harmonic emission is a coherent many-electron observable, so an edge state need not influence the total radiation in the same way that it changes a one-electron spectrum.

Research problem

Using the continuous-space finite-chain model below, determine whether and how the chain's dimerization topology is encoded in its low-energy high-harmonic response. Establish what role is played by edge-state occupation, and distinguish it from an explanation based only on the static gap, the structural distortion, or the time dependence of the self-consistent Kohn-Sham potential.

The derivation, numerical construction, choice of discriminating evidence and organization of the argument are yours. Establish the mechanism from your calculation rather than assuming that the presence of an in-gap state is by itself sufficient.

Physical inputs

Use atomic units. The chain contains $N=100$ fixed singly charged ions and $N=100$ electrons in the spin-neutral configuration $N_\uparrow=N_\downarrow=50$. With $a=2$, define

$$x_i=(i-\frac{N+1}{2})a,x_i^{\mathrm{\prime }}(\delta )=x_i-(-1{)}^i\delta ,i=1,\dots ,N,x\_i=\backslash left(i-\backslash frac\{N+1\}\{2\}\backslash right)a,\; \backslash qquad\; x\_i\text{'}(\backslash delta)=x\_i-(-1)^i\backslash delta,\; \backslash qquad\; i=1,\backslash ldots,N,$$

and

$$v_{\mathrm{i}\mathrm{o}\mathrm{n}}(x)=-\sum _{i=1}^N\frac{1}{\sqrt{[x-x_i^{\mathrm{\prime }}(\delta ){]}^2+1}}\mathrm{.}v\_\{\backslash mathrm\{ion\}\}(x)=-\backslash sum\_\{i=1\}^\{N\}\; \backslash frac\{1\}\{\backslash sqrt\{[x-x\_i\text{'}(\backslash delta)]^2+1\}\}.$$

Use the continuous-space, self-consistent x-only local-spin-density model with the same soft-core interaction in the Hartree and ion-ion terms and

$$v_{\mathrm{x}\mathrm{c},\sigma }(x,t)=-{[\frac{6}{\pi }{n}_{\sigma }(x,t)]}^{1\mathrm{/}3}\mathrm{.}v\_\{\backslash mathrm\{xc\},\backslash sigma\}(x,t)=-\; \backslash left[\backslash frac\{6\}\{\backslash pi\}n\_\backslash sigma(x,t)\backslash right]^\{1/3\}.$$

Correlation is neglected. An SSH or other tight-binding Hamiltonian is not a substitute for this physical model.

Investigate the following fixed geometries:

Task label	Dimerization shift
M	$\delta=0$
D+	$\delta=0.265$
D-	$\delta=-0.165$

These shifts are prescribed structural inputs for the comparison; they are not stated here as numerical minima of a student-computed energy curve.

Drive each geometry with

and $A(t)=0$ otherwise, using

$$A_0=0.1,\omega =0.0075,n_{\mathrm{c}\mathrm{y}\mathrm{c}}=5.A\_0=0.1,\; \backslash qquad\; \backslash omega=0.0075,\; \backslash qquad\; n\_\{\backslash mathrm\{cyc\}\}=5.$$

The field is polarized along the chain. Construct the graded radiation observable consistently from the spin-summed total dipole and report it against harmonic order. Declare the spectral transform and any window used. Use one common constant display factor across every case or dynamical comparison for which relative magnitude is scientific evidence. Absolute magnitude is not a grading target.

Method boundary

Use standard stationary and time-dependent Kohn-Sham methods for the stated continuous-space functional. Initialize the driven dynamics from the corresponding stationary occupied orbitals. The comparison or intervention used to distinguish static structure from the effect of the evolving self-consistent Kohn-Sham potential is yours.

All task-specific analytical, numerical and evidential constructions are yours. No spatial or temporal grid, box, boundary treatment, eigensolver, propagator, number of displayed orbitals, spectral floor, plotting range, diagnostic order or validation sequence is prescribed. You are responsible for checking that the reported evidence is sufficiently reliable and reproducible for the conclusions made. Numerical reliability has no separate score and is reflected in the correctness and quality of the physical evidence.

Submission

Submit one self-contained scientific report and an offline reproduction package containing the derivation, executable code, reconstructible numerical data underlying every central structural and radiation claim, references actually used, and concise run and environment information. The organization and number of figures are yours. No fixed file tree, filename scheme, complete exploratory history or examiner-selected control is required.

Include a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge.

The active examination resources and limits are defined in the companion resource_limits.json. All retained production calculations must run within that fixed allocation and active wall time. More expensive modelling does not receive additional credit.

Reference and integrity policy

This examination is conducted without network access or live literature retrieval. You may rely on your existing general scientific knowledge and, if any are explicitly supplied with the examination, 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 during the examination.

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.