# 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

$$
x_i=\left(i-\frac{N+1}{2}\right)a,
\qquad
x_i'(\delta)=x_i-(-1)^i\delta,
\qquad i=1,\ldots,N,
$$

and

$$
v_{\mathrm{ion}}(x)=-\sum_{i=1}^{N}
\frac{1}{\sqrt{[x-x_i'(\delta)]^2+1}}.
$$

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

$$
v_{\mathrm{xc},\sigma}(x,t)=-
\left[\frac{6}{\pi}n_\sigma(x,t)\right]^{1/3}.
$$

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

$$
A(t)=A_0\sin^2\!\left(\frac{\omega t}{2n_{\mathrm{cyc}}}\right)
\sin(\omega t),
\qquad 0<t<\frac{2\pi n_{\mathrm{cyc}}}{\omega},
$$

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

$$
A_0=0.1,
\qquad \omega=0.0075,
\qquad n_{\mathrm{cyc}}=5.
$$

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