cases/case05/sources.md

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

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 $$n_{\sigma }(x,t)=\sum _{i\in \mathrm{o}\mathrm{c}\mathrm{c}}\mathrm{\mid }{\phi }_{\sigma i}(x,t){\mathrm{\mid }}^2,u[n](x,t)=\int \frac{n(x^{\mathrm{\prime }},t)d{x}^{\mathrm{\prime }}}{\sqrt{(x-x^{\mathrm{\prime }}{)}^2+1}},v_{x,\sigma }=-[6n_{\sigma }\mathrm{/}\pi {]}^{1\mathrm{/}3}\mathrm{.}n\_\backslash sigma(x,t)=\backslash sum\_\{i\backslash in\backslash mathrm\{occ\}\}|\backslash varphi\_\{\backslash sigma\; i\}(x,t)|^2,\backslash quad\; u[n](x,t)=\backslash int\backslash frac\{n(x\text{'},t)dx\text{'}\}\{\backslash sqrt\{(x-x\text{'})^2+1\}\},\backslash quad\; v\_\{x,\backslash sigma\}=-[6n\_\backslash sigma/\backslash pi]^\{1/3\}.$$ The coherent observable is $$X(t)=\sum _{i,\sigma }\int x\mathrm{\mid }{\phi }_{\sigma i}(x,t){\mathrm{\mid }}^{2}dx,S_D(\mathrm{\Omega })\propto \mathrm{\mid }\mathcal{F}[wX](\mathrm{\Omega }){\mathrm{\mid }}^2,X(t)=\backslash sum\_\{i,\backslash sigma\}\backslash int\; x|\backslash varphi\_\{\backslash sigma\; i\}(x,t)|^2dx,\backslash qquad\; S\_D(\backslash Omega)\backslash propto|\backslash mathcal\; F[wX](\backslash Omega)|^2,$$ 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.