# Scientific Sources — Case05

## References

Dieter Bauer and Kenneth K. Hansen. 2018. *High-Harmonic Generation in Solids with and without Topological Edge States*. *Physical Review Letters*, 120, 177401. [Source](https://doi.org/10.1103/PhysRevLett.120.177401)

High-Harmonic Generation in Solids with and without Topological Edge States

## Criterion-to-source crosswalk

Article page identifiers below are journal pages, not PDF viewer indices. Supplemental pages are explicitly marked. The crosswalk distinguishes direct source facts, mathematical entailments and task-defined evidence duties. Original images and PDFs are referenced only; none are reproduced.

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Bauer and Hansen (2018), article p.177401-2, Eqs.(1)-(6): soft-core continuous-space x-only LSD chain, spin-neutral occupation and fixed dimerized geometries; not a tight-binding replacement. |
| F2, O1, I1, M1 | Bauer and Hansen (2018), article pp.177401-2-3, Fig.1(b-d), Eq.(6) and edge-state paragraph: half-filled metal, dimerized insulators and an in-gap edge pair with only its lower member occupied. Fig.1(a) exact energy minima are not compulsory targets. |
| F3, F4 | Bauer and Hansen (2018), article p.177401-3, Eqs.(7),(8): driven KS orbitals, spin-summed coherent total dipole and squared Fourier modulus; current/acceleration variants require omega^2/omega^4 weighting. |
| O2, I2, M2 | Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and sub-gap discussion: large D- (source phase B) / D+ (source phase A) contrast on common scale; exact 14-order magnitude or colors are not targets. |
| O3, I3, M6 | Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and frozen-KS/screening paragraphs: insulating contrast survives frozen internal potential, while metal is much more sensitive. A functionally equivalent intervention is accepted. |
| O4, I4, M3, M4, M5 | Bauer and Hansen (2018), article p.177401-4, Fig.3 and paragraph beginning "The key question": incoherent orbital powers versus coherent total distinguish cancellation. The same page, "Topological vs other edge states", describes impurity/occupation controls and a virtual extra electron filling the vacant edge state; these are optional canonical causal examples. |
| M7, V1 | Bauer and Hansen (2018), article pp.177401-2-4, model equations and conclusion delimit finite-chain mean-field scope. Exact Fig.1(a) energy-curve reproduction is excluded by the task/effective rubric because its printed-functional reproduction remains unresolved; risk validation is task-defined. |

## Formula and convention notes

For F1/F3-F4/M3-M4, article pp.177401-2-4, Eqs.(3),(4),(8) and Fig.3, the spin densities and internal potential obey
$$n_\sigma(x,t)=\sum_{i\in\mathrm{occ}}|\varphi_{\sigma i}(x,t)|^2,\quad
u[n](x,t)=\int\frac{n(x',t)dx'}{\sqrt{(x-x')^2+1}},\quad
v_{x,\sigma}=-[6n_\sigma/\pi]^{1/3}.$$
The coherent observable is
$$X(t)=\sum_{i,\sigma}\int x|\varphi_{\sigma i}(x,t)|^2dx,\qquad
S_D(\Omega)\propto|\mathcal F[wX](\Omega)|^2,$$
not $\sum_{i,\sigma}|\mathcal F[wx_{\sigma i}]|^2$. The latter is an optional cancellation diagnostic. Derivative-form spectra differ by $\Omega^2$ or $\Omega^4$ when transform/window boundary terms are controlled. Frozen-KS comparison freezes the internal Hartree/exchange potential at its stationary value, not the external laser coupling.

The continuous-space orbital equation is $i\partial_t\varphi_{\sigma i}=[-\partial_x^2/2+v_{\rm ion}(x)-iA(t)\partial_x+u[n]+v_{x,\sigma}]\varphi_{\sigma i}$, with $v_{\rm ion}(x)=-\sum_j[(x-x_j)^2+1]^{-1/2}$ and $x_j=(j-(N+1)/2)a-(-1)^j\delta$. This is the x-only LSD velocity-gauge convention of source Eqs.(1)-(6); a common time-dependent scalar A-squared phase does not change the graded density/dipole.

## Scope of the source anchors

The locators identify scientific objects and mechanisms, not compulsory figure layouts, solver settings, exact grids, pointwise tolerances or a fixed evidence package. Canonical examples may be replaced by functionally equivalent evidence subject to the rubric. Source readback verifies the candidate's science; it cannot supply missing candidate presentation, identification or inference. Reporting, applicability/N/A and descriptive Research Closure duties remain those stated in the rubric and task.