Yang–Baxter Structures & Bethe Ansatz

Provide the reusable algebraic and spectral machinery used across quantum chains, statistical models, stochastic models and field theories.

The two-magnon Bethe derivation gives a concrete starting point: match the contact equation, impose the periodic boundary, and check that the resulting vector is nonzero. The XXX learning sequence develops the construction step by step and tests selected states against an independently built Hamiltonian.

The commuting-transfer-matrix derivation develops the algebraic structure: the rational Yang–Baxter identity, RTT relation, commuting traces and extraction of the XXX Hamiltonian. A four-lesson learning route supplies intermediate practice, regular Bethe-vector construction and finite-matrix checks. Constructing these commuting operators does not by itself prove independence of all charges or completeness of a Bethe parametrization. The chapter map identifies broader planned coverage, with available readings linked explicitly.

For a concrete completeness proof, A complete four-site XXX sector constructs six orthonormal states from regular roots, a physical singular pair and symmetry descendants. Their projectors resolve the identity in that sector, showing why a correct list of energies can still miss states. The proof’s finite scope is explicit.