Mathematics

The backbone the physics leans on. Thin by design — these exist so the physics notes can point here instead of re-deriving. The Atlas places them in the ladder.

Many-body formalism

• Second quantization L0 🌳 — Fock space & occupation numbers, CAR/CCR, field operators, one-/two-body operators, normal ordering & Wick’s theorem

Band theory

• Tight-binding & Bloch’s theorem L0 🌳 — cell-periodic $u_{n\mathbf{k}}$, the LCAO secular equation $H(\mathbf{k})c=ES(\mathbf{k})c$, and the dynamical-matrix analogy
• Maximally-localized Wannier functions L2 🌳 — $U(\mathbf{k})$ gauge freedom, the spread functional $\Omega ={\Omega }_I+\tilde{\Omega }$, disentanglement, Wannier interpolation

Symmetry

• Crystal symmetry & space groups L0 🌳 — the little group & star of $\mathbf{k}$, its irreps and compatibility relations, Wyckoff site symmetry, and the symmetry-adapted normal-mode decomposition

Linear algebra for many-body problems

• BdG & para-Hermitian diagonalization L1 🌳 — the para-Hermitian metric $g$, Colpa’s Cholesky route, stability ↔ positive-definiteness

Green’s functions

• Green’s functions & LKAG L1 🌳 — Lehmann/spectral $A(\omega )$, self-energy $\Sigma$ & the Dyson equation, quasiparticle weight $Z$, the LKAG exchange $J_{ij}$