Wolfbook

A chain of $L$ spins-$\frac{1}{2}$ with nearest-neighbour exchange $H={\sum }_i{\vec{S}}_i\cdot {\vec{S}}_{i+1}$, exactly solvable by the Bethe Ansatz. A twisted boundary condition $S_{L+1}^{\pm }=e^{\pm i\varphi }{S}_1^{\pm }$ interpolates between periodic and antiperiodic BC — a clean probe of integrability.

siteOp[op,k,L] (KroneckerProduct embedding) + Hham[L_,phi_] builds the full $2^L\times 2^L$ matrix. Spectra for $L=2,3,4$ swept over $\varphi \in [0,2\pi ]$ (80 steps). Then NNN deformation $H_{\mathrm{NI}}=H+\delta \sum {\vec{S}}_i\cdot {\vec{S}}_{i+2}$ ($\delta =0.3$): integrable shows smooth level crossings; non-integrable shows avoided crossings.

Artifacts: spinchain.wb (12 cells) · spinchain_report.tex

The symmetry algebra of the $\mathcal{N}=4$ SYM spin chain (Beisert 2006). Has 14 generators: ${\mathbf{L}}^a{}_b$, ${\mathbf{R}}^{\alpha }{}_{\beta }$ (bosonic) + ${\mathbf{Q}}^a{}_{\alpha }$, ${\mathbf{S}}^{\alpha }{}_a$ (fermionic). Pure $\mathfrak{psu}(2|2)$ does not close — Jacobi identities force three central charges $\mathbf{C},\mathbf{P},\mathbf{K}$.

NCAlgebra loaded; ${\mathbb{Z}}_2$ grading set; super-bracket $[A,B\}=AB-(-1{)}^{|A||B|}BA$ implemented as GB0[A,B]. Systematic Jacobi check for all generator triples — 7 identities fail without central terms. Adding $\{\mathbf{C},\mathbf{P},\mathbf{K}\}$ to $\{\mathbf{Q},\mathbf{S}\}$ anticommutator: all close.

Artifacts: psu22.wb (multi-section) · psu22_report.tex

Scalar QED: $\mathcal{L}=(D_{\mu }\varphi {)}^{†}{D}^{\mu }\varphi -m^2|\varphi {|}^2-\frac{1}{4}{F}_{\mu \nu }^2$. Compton scattering $\gamma \varphi \to \gamma \varphi$ has 3 tree-level (seagull, s-, u-channel) and 53 one-loop diagrams. Agent installed FeynArts 3.12 and wrote the model file autonomously.

Custom ScalarQED.mod model file (SSV 3-point + SSVV seagull vertices). CreateTopologies + InsertFields for $2\to 2$ scattering. Full 53-diagram 1-loop sheet exported as PNG/PDF; diagrams further split into topology classes (self-energy, vertex, box) with DiagramExtract.

Artifacts: Fyanman.wb · ComptonScalarQED_notes.tex