Quantum Integrability

Learn finite quantum chains through magnon scattering, Bethe equations and commuting transfer matrices, with preparation in complex linear algebra.

For preparation, use Eigenvalues, commutators & tensor products to check complex norms, ordered tensor bases and invariant sectors with explicit small matrices.

Begin with the XXX spin-chain sequence: build the Hamiltonian, solve one magnon, derive two-magnon scattering and ring quantization, then check a Bethe eigenvector against exact diagonalization. The six steps have worked examples, practice and solutions.

Then follow four lessons on the algebraic mechanism: verify a rational R-matrix, build monodromy and transfer matrices, derive commuting charges and the Hamiltonian, then construct regular Bethe eigenvectors. You can enter this sequence directly if tensor products and the spin-chain Hamiltonian are already familiar.

Use that state in two lessons on observables: normalize a Bethe vector, calculate spin matrix elements, and reconstruct an exact finite-chain correlation. Selection rules and sum rules expose mistakes that an energy spectrum alone cannot detect. The chapter map links these sequences and describes further planned coverage, including thermodynamics.