Physics

Condensed-matter physics and many-body methods. Organized by sub-area below; the depth of each note is in its level/* tag, and The Atlas shows the whole ladder.

Foundations

• Born–Oppenheimer PES L0 🌳 — the full molecular Hamiltonian, adiabatic separation by mass ratio, and the PES as the root of the phonon branch

Magnetism & spin dynamics

• Linear spin-wave theory L2 🌳 — Holstein–Primakoff, the bosonic Hamiltonian, Colpa diagonalization
• Dzyaloshinskii–Moriya & exchange anisotropy L1 🌳 — the antisymmetric ${\mathbf{D}}_{ij}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)$; Moriya’s SOC mechanism & symmetry rules; canting/spiral/skyrmion stabilization
• Exchange tensors from broken-symmetry DFT L2 🌳 — decomposing ${\mathbf{J}}_{ij}$ into isotropic $J$ / DMI $\mathbf{D}$ / anisotropic $\mathbf{\Gamma }$; LKAG (TB2J) vs. four-state energy mapping; the SOC/+U caveats
• Mermin–Wagner L1 🌳 — no continuous-symmetry order at $T>0$ in $d\le 2$; the magnon IR divergence; anisotropy & dipolar loopholes; Ising/BKT contrasts

Lattice dynamics

• The dynamical matrix & the acoustic sum rule L1 🌳 — $D_{\alpha \beta }^{\kappa {\kappa }^{\prime }}(\mathbf{q})$, the ${\omega }^2$ eigenproblem, the translational ASR, the LO–TO non-analytic term
• Anharmonic phonons — QHA, SCHA & the SSCHA L2 🌳 — cubic/quartic FCs, QHA vs SSCHA renormalization, the free-energy Hessian, finite-$T$ soft-mode stabilization
• Electron–phonon coupling L2 🌳 — $g_{mn\nu }(\mathbf{k},\mathbf{q})$, ${\alpha }^{2}F$ & $\lambda$, Allen–Heine–Cardona gap renormalization, phonon-limited mobility, Fröhlich + Wannier/EPW
• Chiral phonons in tellurium L3 🌿 — phonon angular momentum, pseudo-angular-momentum selection rules, chiral space groups $P{3}_{1}21$/$P{3}_{2}21$
• Imaginary modes: artifact vs. instability L2 🌳 — (filed under methods; it’s a workflow diagnostic)

Berry phase & topology

• Berry phase & the modern theory of polarization L2 🌳 — Berry connection/curvature, KS–Vanderbilt polarization, the polarization quantum, $Z^{\ast }$ as a Berry-phase derivative, gateway to the Chern number

Electronic structure / methods

• See Methods for DC-DFT, +U, hybrids, and the self-interaction picture.