Using stripe order in the two-dimensional Hubbard model as an example, connect the current Codex Skills into a condensed-matter research workflow, from choosing a question and reviewing literature to derivations, numerical experiments, paper delivery, and reusable procedures.
Starting from Hamilton’s equations, understand why the symplectic form captures the structure of classical dynamics more directly than coordinates.
View the action as a generating function and understand how families of classical trajectories become geometric objects in phase space.
How symmetries generate conserved quantities on symplectic manifolds, and why Lie group actions provide the right framework.
How the stationary-phase approximation reorganizes quantum amplitudes around classical trajectories.
From correlation length to universality, build geometric intuition for continuous phase transitions.