The Atlas

What this is

The whole GRID as one tree, ordered by depth instead of topic. Within each area, notes climb from foundations to the research frontier; ”←” marks a prerequisite edge. This is the view the graph can’t give you — link topology isn’t the same as a learning order.

Legend

Levels — L0 foundations · L1 core working knowledge · L2 advanced / what I actually use · L3 open frontier. Defined in conventions. Maturity — 🌱 seedling (sparse, maybe wrong) · 🌿 growing · 🌳 evergreen (reasonably complete).

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Physics — index

L0 — foundations

• 🌳 Born–Oppenheimer PES — $H=T_n+T_e+V_{ne}+V_{ee}+V_{nn}$, adiabatic separation in $\kappa =(m_e/M{)}^{1/4}$, the PES $E(\mathbf{R})=E_{\text{elec}}+V_{nn}$ whose mass-weighted Hessian is the dynamical matrix; BO breakdown at conical intersections ← prereqs foundational (many-electron Schrödinger eq. · Rayleigh–Ritz), written as prose

L1 — core

• 🌳 Dynamical matrix & the acoustic sum rule — $D_{\alpha \beta }^{\kappa {\kappa }^{\prime }}(\mathbf{q})$ as the mass-weighted FT of $\Phi$, the ${\omega }^2$ eigenproblem, the translational ASR & its numerical enforcement, the LO–TO non-analytic term ← BO-PES, HF forces, crystal symmetry
• 🌳 Dzyaloshinskii–Moriya & exchange anisotropy — the antisymmetric ${\mathbf{D}}_{ij}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)$: Dzyaloshinskii’s Landau invariant, Moriya’s first-order-in-SOC mechanism & symmetry rules ($\mathbf{D}=0$ at an inversion-centred bond); canting → spirals → skyrmions ← the Heisenberg/super-exchange Hamiltonian & spin–orbit coupling (prose)
• 🌳 Mermin–Wagner — no spontaneous continuous-symmetry order at $T>0$ for short-range $d\le 2$; the Bogoliubov-inequality / $\int d^{2}k/k^2$ magnon IR divergence; anisotropy & dipolar loopholes; Ising vs. BKT contrasts ← the Heisenberg model (prose)

L2 — advanced

• 🌳 Linear spin-wave theory — Holstein–Primakoff → BdG → Colpa ← DMI, BdG algebra
• 🌳 Exchange tensors from broken-symmetry DFT — ${\mathbf{J}}_{ij}=J1+\mathbf{D}+\mathbf{\Gamma }$ from DFT: LKAG/Green’s-function force theorem (TB2J) vs. four-state energy mapping; SOC turns on $\mathbf{D},\mathbf{\Gamma }$; the +U/projector/supercell caveats ← Green’s functions, DMI, +U, magnetic anisotropy (prose)
• 🌳 Berry phase & the modern theory of polarization — polarization as a Berry phase mod the quantum $e\mathbf{R}/\Omega$; $Z^{\ast }$ from the polarization side; gateway to topology ← tight-binding & Bloch, Wannier functions
• 🌳 Anharmonic phonons — QHA, SCHA & the SSCHA — QHA vs SSCHA; the free-energy Hessian (not $\mathbf{\Phi }$); finite-$T$ soft-mode stabilization ← dynamical matrix, imaginary modes
• 🌳 Electron–phonon coupling — $g_{mn\nu }(\mathbf{k},\mathbf{q})$, ${\alpha }^{2}F$ & $\lambda$, Allen–Heine–Cardona, Fröhlich + Wannier/EPW ← dynamical matrix, DFPT, Wannier functions

L3 — frontier

• 🌿 Chiral phonons in tellurium — phonon angular momentum & PAM selection rules; chiral space groups $P{3}_{1}21$/$P{3}_{2}21$ ← dynamical matrix, crystal symmetry

Methods — index

L0 — foundations

• 🌳 Kohn–Sham DFT — Hohenberg–Kohn theorems, the non-interacting KS mapping, $F=T_s+E_H+E_{xc}$ defining $E_{xc}$; small-but-decisive xc energy ← Born–Oppenheimer PES, second quantization

L1 — core

• 🌳 Hartree–Fock theory — best single determinant; $\hat{J}/\hat{K}$, exact cancellation of the Hartree self-interaction; Koopmans; Roothaan–Hall ← Born–Oppenheimer PES, second quantization
• 🌳 XC functionals & Jacob’s ladder — LSDA→GGA→meta-GGA (PBE, SCAN) anchored by exact constraints; semilocal-accuracy axis orthogonal to the SIE axis ← Kohn–Sham DFT, self-interaction error
• 🌳 DFT+U — the Hubbard-$U$ correction: Dudarev $U_{\mathrm{eff}}$ vs. full Liechtenstein $U,J$; the occupation matrix & projector ambiguity; FLL vs. AMF double counting; non-universal $U$ from linear response / cRPA ← Kohn–Sham DFT, XC & Jacob’s ladder
• 🌳 Hybrid functionals — adiabatic-connection rationale; PBE0 ($\frac{1}{4}$ EXX) vs. HSE range separation ($\mathrm{erfc}/\mathrm{erf}$, $\omega$); gaps as partial SIE cancellation; the Fock-exchange / $k$-mesh cost ← XC & Jacob’s ladder, Hartree–Fock
• 🌳 Phonons by finite displacement — the frozen-phonon supercell route: symmetry-reduced displacements, the supercell-vs-range tradeoff, ASR/FC_SYMMETRY, the phonopy + CRYSTAL23 workflow; contrasted with DFPT ← dynamical matrix, HF forces, crystal symmetry
• 🌳 Hellmann–Feynman forces & the Pulay term — forces as the BO-PES gradient; electrostatic force theorem; Pulay term for atom-centred bases corrupts phonons ← Born–Oppenheimer PES
• 🌳 Self-interaction & delocalization error — one- vs many-electron SIE; the convexity / flat-plane condition that +U, hybrids, DC-DFT all chase
• 🌳 The beyond-DFT ladder — which error each rung fixes; honest on +U-as-SIE-patch and FLL vs AMF ← SIE, +U, hybrids, DC-DFT

L2 — advanced

• 🌱 Density-driven error (DC-DFT) — splitting functional from density error; HF densities; the forces/MLIP angle
• 🌳 Imaginary modes: artifact vs. instability — soft-mode artifact vs. true instability: ASR/basis-set artifacts, and the Born–Huang rotational + Huang vanishing-stress conditions for the 2D flexural (ZA) branch; worked through on monolayer SnSe₂ ← finite displacement, dynamical matrix & ASR
• 🌳 DFPT & linear response — Sternheimer, the 2n+1 theorem, $Z^{\ast }$ + ${\epsilon }_{\infty }$ as the LO–TO inputs ← dynamical matrix
• 🌳 Machine-learned interatomic potentials — MACE/NequIP vs GAP/ACE; force/stress training; the DC-DFT data caveat ← DC-DFT
• 🌿 Phonons & anharmonicity from MLIPs — force constants from MLIPs, anharmonic sampling, density-driven poisoning of FCs ← MLIPs, finite displacement, DC-DFT
• 🌳 GW & quasiparticles — $\Sigma =iGW$, $G_0{W}_0$ vs QSGW, the derivative-discontinuity gap; DMFT as context ← Green’s functions

Mathematics — index

L0 — foundations

• 🌳 Second quantization — Fock space & occupation numbers; fermionic CAR vs bosonic CCR; field operators; one-/two-body operators; normal ordering & Wick’s theorem
• 🌳 Crystal symmetry & space groups — the little group & star of $\mathbf{k}$, irreps and compatibility relations, Wyckoff site symmetry, and the symmetry-adapted normal-mode decomposition; the Herring test
• 🌳 Tight-binding & Bloch’s theorem — cell-periodic $u_{n\mathbf{k}}$, the LCAO secular equation $H(\mathbf{k})c=ES(\mathbf{k})c$, and the dynamical-matrix analogy

L1 — core

• 🌳 BdG & para-Hermitian diagonalization — the para-Hermitian metric $g$, why naive Hermitian diagonalization fails, Colpa’s Cholesky route; positive-definiteness as the stability test ← second quantization
• 🌳 Green’s functions & LKAG — Lehmann/spectral $A(\omega )$, the Dyson equation & self-energy $\Sigma$, quasiparticle weight $Z$ vs. incoherent background, Matsubara; the LKAG exchange $J_{ij}$ ← second quantization

L2 — advanced

• 🌳 Maximally-localized Wannier functions — $U(\mathbf{k})$ gauge, $\Omega ={\Omega }_I+\tilde{\Omega }$ (Marzari–Vanderbilt), disentanglement, Wannier interpolation ← tight-binding & Bloch, crystal symmetry

Materials & systems — index

L1 — core

• 🌳 Perovskite structure (ABX₃) — corner-sharing $B{X}_6$ aristotype; Goldschmidt $t$ & octahedral $\mu$; Glazer tilts; cubic→tetragonal→orthorhombic soft-mode transitions ← crystal symmetry

L2 — advanced

• 🌱 WSe₂ heterostructure — the phonon campaign, and the W-basis cautionary tale
• 🌱 CrI₃ and CrSBr — 2D FM vs. A-type AFM; a documented DFT-failure target
• 🌿 Perovskite variants — double & antiperovskites — $A_{2}B{B}^{\prime }{X}_6$ rock-salt B-site order ($Fm\bar{3}m$) & $X_{3}BA$ site inversion; Sr₂FeMoO₆, Mn₃GaN, Li₃OCl ← perovskite structure, crystal symmetry

L3 — frontier

• 🌱 Flat-band & Kekulé models — α–T₃, flat-band self-interaction error as the systematic