Scientific sources - Case42
References
Daemo Kang, Tien-Tien Yeh, Takahiro Morimoto, and Alexander V. Balatsky. 2025. Kapitza-Dirac interference of Higgs waves in superconductors. arXiv, 2511.10954. Source. Version 1.
Inertial complex TDGL S/P responses and evidence-bounded classification. Single-beam is Fig. 3; strong two-beam is Fig. 2; weak Fig. 5 is optional discrimination, not a compulsory third branch. No absolute source wavevector or fixed-vortex-count target is imposed.
Criterion-to-source crosswalk
| Criteria | Source locations, scientific use and limits |
|---|---|
| F1, F2 | Kang et al. (2025), article pp. 2–3, Eqs. (1)–(3), initial-state and boundary paragraphs. Inertial complex-field initial-value problem and propagation. Peak normalization and far-field ground-state boundary identity are explicitly task-defined, not guessed from the source Dirichlet label. |
| O1, I1 | Article p. 5, Fig. 3(a), caption, supercurrent/vorticity paragraph; Eqs. (4)–(6), p. 3. Single Gaussian S branch, co-located depletion and phase/topological organization. Winding, covariant current or equivalent complex-field evidence is accepted without a mandatory triad or fixed count. |
| O2 | Article p. 5, Fig. 3(b,c) and Higgs/NG paragraph; transform definition on p. 3. Complementary collective response for the same S field, not another real-space amplitude image. |
| O3, I2 | Article p. 4, Fig. 2(a), caption and opening Results paragraphs; Eq. (6), p. 3. Strong two-beam P complex/topological real-space response, not the weak two-beam Fig. 5 case. |
| O4, I3 | Article p. 4, Fig. 2(b,c), opening Results; pp. 6–7, Fig. 5(b,c) and contrast paragraph. Complementary P evidence must carry material classification information. Orthogonal momentum cuts discriminate the canonical example but are not an exclusive required display. |
| I4 | Article pp. 4–5, Figs. 2(b,c), 3(b,c), frequency-family discussion. Candidate identification of near-drive/phase and near-Higgs/amplitude families; identification is distinct from a causal mechanism or source attribution. |
| M1, M2 | Article p. 3, Eq. (3); p. 5, Eq. (7) and surrounding coupling paragraphs. Physical expansion of the covariant kinetic term distinguishes quadratic-A amplitude driving from linear-A/current/phase-gradient coupling. |
| M3 | Article p. 5, Fig. 3(a–c), Eq. (7) and single-beam vortex/Higgs discussion. Positive causal link between S phase/amplitude organization and collective response, beyond coexistence. |
| M4, M5 | Article pp. 4–5, Fig. 2(a–c), Eq. (7), vortex-mediated linear-coupling discussion. Positive P topology/phase-gradient mechanism and its collective signature. A missing preferred control cannot substitute for judging this mechanism. |
| M6 | Article pp. 4–7, Figs. 2(a–c), 3(a–c), 5(a–c), associated directional comparison. Task-defined evidence-bounded physical classification, including a correct supported negative or restricted judgment; a source-preferred label is not compulsory. |
| M7 | Article p. 4, Fig. 2 caption and Results parameters; p. 5, Fig. 3 caption. S/P change beam count, amplitude, waist and damping together; physical entailment forbids treating them as a one-variable causal control. |
| V1 | Article p. 3, Eq. (3), initial/boundary and transform definitions; Figs. 2(b,c), 3(b,c), pp. 4–5. Task-defined conditional risk against boundary, switch-on, transform and coordinate artefacts. No mandatory grid, late-time window or absolute source momentum peak. |
| V2 | Article pp. 5–7, weak-two-beam discussion and Fig. 5(a–c). Ordinary interference is a possible alternative; the weak source case is optional. Same-object control, localization or equivalent evidence may discriminate it; honest unresolved classification remains possible. |
| V3 | Article pp. 2–3, phenomenological model; pp. 5–7, mode-observability and interpretation caveats. Fixed-A two-dimensional model and computed-case scope. Absolute printed momentum coordinates and exact vortex totals are not quantitative gold. |
Formula and convention notes
In the stated dimensionless model, $\tau\partial_t^2\psi+\gamma\partial_t\psi=\psi-|\psi|^2\psi+(\nabla-i\mathbf A)^2\psi$, $\tau=1/2$, $\psi(0)=1$, $\partial_t\psi(0)=0$ (Eq. (3), task initial-velocity identity). The Gaussian member uses $\mathbf A=\hat{\mathbf x}\cos t\sum_jA_j\exp[-((x-x_j)^2+y^2)/w_0^2]$ with task-declared per-beam peak amplitudes, not an additional hidden LG normalization.
The covariant source expands as $(\nabla-i\mathbf A)^2\psi=\nabla^2\psi-2i\mathbf A\cdot\nabla\psi-i(\nabla\cdot\mathbf A)\psi-A^2\psi$. For $\psi=\rho e^{i\theta}$, $i\mathbf A\cdot\nabla\psi=i\mathbf A e^{i\theta}\cdot\nabla\rho-\rho e^{i\theta}\mathbf A\cdot\nabla\theta$ (Eq. (7)). These are positive amplitude/phase-coupling mechanisms; neither coexistence alone nor alternative exclusion alone establishes them.
$\mathbf J_s=\operatorname{Im}[\psi^*(\nabla-i\mathbf A)\psi]=\rho^2(\nabla\theta-\mathbf A)$ and $\omega_s=\nabla\times\mathbf J_s$ are source-defined diagnostics. A phase winding $w=(2\pi)^{-1}\oint\nabla\theta\cdot d\boldsymbol\ell$ around a contour avoiding zeros is a derived equivalent topological test. Amplitude minima alone do not establish winding; no prescribed topology triad is required.
With one declared convention $\widetilde{\delta\psi}(\mathbf k,\Omega)=\int d^2r,dt,[\psi(\mathbf r,t)-1]e^{-i\mathbf k\cdot\mathbf r+i\Omega t}$; equivalent background/normalization choices are accepted if they preserve the claim-bearing family. Angular wavevector $k$ and cyclic wavevector $q=k/(2\pi)$ must not be silently interchanged. The source printed coordinate ambiguity prevents exact absolute-peak grading.
Scope of the source anchors
Source examples verify physical identity and robust structure; their numerical samples, plotting conventions and panel layout are not compulsory candidate deliverables. Reporting, reconstructibility, evidence attribution, conditional validation and Research Closure are examination-defined obligations, not additional results asserted by the article. A derivation or physical entailment is identified as such rather than presented as a directly printed source formula.