Methods

Exact & non-interacting

• Tight binding Diagonalising the non-interacting lattice in momentum space — bands, Fermi surfaces, density of states. 1 topic

Planned Bethe ansatzExact diagonalization

Perturbative & diagrammatic

• Perturbation theory Controlled expansions in a small parameter — here, hopping over interaction, giving superexchange. 1 topic

Planned Bethe–Salpeter / channelsDiagrammaticsRandom phase approximation

Mean-field & decoupling

• Mean-field theory Decoupling interactions into a self-consistent background field; the first window onto broken symmetry. 1 topic

Planned BCS / pairing mean fieldHartree–FockLandau–Ginzburg theory

Field-theoretic

• Green's functions Single- and many-particle propagators — the central objects that carry spectra, lifetimes, and response. 1 topic

Planned Luttinger–Ward / Baym–KadanoffPath integral

Renormalization & effective theories

Planned Effective theoriesRenormalization group

Quantum embedding (DMFT family)

• Dynamical mean-field theory Mapping the lattice onto a self-consistent quantum impurity — exact in infinite dimensions, the workhorse for strong correlation. 2 topics

Planned Cluster DMFT (CDMFT)Cluster perturbation theoryVariational cluster approximation

Impurity & Monte Carlo solvers

• Iterated perturbation theory A cheap second-order solver for the DMFT impurity problem — transparent, and surprisingly faithful at half filling. 1 topic

Planned Continuous-time QMCDeterminant QMC

Tensor networks

Planned DMRGTensor networks

Spin & semiclassical

Planned Large-N / large-SSpin-wave theory

Symmetry & topology

Planned Group theoryTopological methods

Ab initio & downfolding

Planned DFT + DMFTDownfolding

Scaling & approximation

Planned Non-dimensionalizationVariational methods