# Scientific sources - Case26

## References

Candong Liu, Yinghui Zheng, Zhinan Zeng, and Ruxin Li. 2018. *Driving-laser ellipticity dependence of high-order harmonic generation in graphene*. *Physical Review A*, 97, 063412. [Source](https://doi.org/10.1103/PhysRevA.97.063412).

Coherent vector-current ellipticity dependence and model-local channel/optical-coupling mechanism. Robust fifth/seventh/ninth response structures are assessable; exact ninth-order optimum and pointwise component heights are not settled numerical targets.

## Criterion-to-source crosswalk

Locations refer to the article or named preprint/manuscript page, not a figure-file index. Source examples establish scientific identity and robust structure; the task defines required reporting, case coverage and risk-triggered validation. No original source images are reproduced.

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | article pp. 063412-2-3, Eqs. (1)-(3), (5)-(6), (10)-(11), Fig. 1(a,b) Bands and current/dipole matrices derive from the same nearest-neighbor Hamiltonian and complete reciprocal cell. |
| F2 | article pp. 063412-2-4, Eq. (4), Eqs. (12)-(13) and supplied-case paragraph Houston dynamics shifts momentum by A; dephasing acts off diagonal, and the ellipticity normalization preserves the physical drive comparison. |
| F3, F4, M5 | article p. 063412-3, Eqs. (7)-(11) and following spectrum definition; p. 063412-4, yield definition Total current is the coherent sum before squaring, and harmonic yield integrates a peak. Component comparisons preserve a common scale. |
| O1, I1 | article p. 063412-4, Fig. 2(a), HH5 branch, and adjacent text Fifth-order yield is largest for linear driving and is suppressed toward circular driving. |
| O2, O3, I2, M1 | article p. 063412-4, Fig. 2(a), HH7/HH9 branches Higher orders show finite-ellipticity enhancement and circular suppression. The exact ninth-order coherent-total optimum is not a scoring target; source peak coordinates are canonical examples only. |
| O4, I3, M2 | article p. 063412-4, Fig. 3 and interband/intraband discussion The seventh-order interband response carries the robust finite-ellipticity maximum on the stated common scale; equivalent momentum-local discriminators can replace channel plots. |
| M3, M4 | article p. 063412-5, Fig. 4, Eq. (6), and Figs. 5-6 intervention discussion Transition-dipole magnitude/direction near Dirac points gives a microscopic mechanism and intervention examples. No fixed local radius, rotation or source control is required. |
| M6, M7, V1, V2 | article pp. 063412-4-6, model/case discussion, Figs. 2-7; stated task comparison and scope Model-specific robust trends must be separated from representation-sensitive peak/component details. Reliability and triggered-risk treatment do not impose source grids or an exact ninth-order optimum. |

## Formula and convention notes

Source-numbered identities below are paraphrased mathematically. Explicit algebraic consequences and accepted model-conditioned representations are identified as such; they do not add a required solver, plot or validation recipe.

$H_0(\mathbf k)=-\gamma\begin{pmatrix}0&f(\mathbf k)\\f^*(\mathbf k)&0\end{pmatrix}$ gives $E_{c,v}=\pm\gamma|f|$. With common eigenstates, $\mathbf d_{mn}=i\langle u_m|\nabla_k u_n\rangle$ and $\mathbf p_{cv}=i(E_c-E_v)\mathbf d_{cv}$ (Eqs. (1)-(6), (11)); gauge/reciprocal-cell changes must preserve these physical matrices.

At $\mathbf k+\mathbf A(t)$ the Houston density evolves as $\dot\rho=-i[E,\rho]-i\mathbf F\cdot[\mathbf d,\rho]-(1-\delta_{mn})\rho_{mn}/T_2$ (Eq. (4)). The drive components scale as $F_0/\sqrt{1+\epsilon^2}$ and $\epsilon F_0/\sqrt{1+\epsilon^2}$ (Eqs. (12)-(13)), not two independently fixed amplitudes.

Use $\mathbf J=\mathbf J_{\rm intra}+\mathbf J_{\rm inter}$ (Eqs. (7)-(9)) and $I(\omega)\propto\omega^2|\widetilde{\mathbf J}_{\rm intra}+\widetilde{\mathbf J}_{\rm inter}|^2$. The cross term $2\operatorname{Re}(\widetilde{\mathbf J}_{\rm intra}\cdot\widetilde{\mathbf J}_{\rm inter}^*)$ cannot be discarded. $Y_n=\int_{B_n}I(\omega)d\omega$ may use declared equivalent peak bands; independent channel rescaling cannot support a channel-strength comparison.

## Scope of the source anchors

Figure panels, displayed source coordinates, curve multipliers, numerical grids and named controls are reference examples, not compulsory submission layouts or hidden pointwise targets. Equivalent evidence must preserve the actual physical observable, case, relative scale and discriminating function. Correct data, candidate identification, causal explanation and risk resolution are separate evidential acts. Existing branch-local dependencies, applicability/N/A and descriptive Research Closure rules remain in the case rubric.