Classical and quantum many-body systems with as many independent conservation laws as degrees of freedom, permitting exact solution by algebraic rather than perturbative methods. Classical: Liouville-Arnold theorem — N-DOF Hamiltonian…
integrable-systems-physics
XXX Heisenberg spin chain: Bethe-ansatz diagonalisation
Heisenberg 1928 (Z. Physik 49:619) introduced the isotropic XXX Hamiltonian H = -J·Σ S_i·S_{i+1} as a model of ferromagnetism /…
KdV Lax pair: infinitely-many conservation laws via spectral invariance
Lax 1968 (Comm. Pure Appl. Math. 21:467) introduced the paired-operator framework [L, M] = ∂_t L to prove integrability of the KdV…
Yang-Baxter equation + 6-vertex / 8-vertex models
Yang 1967 and Baxter 1972 identified the quartic Yang-Baxter equation (YBE) as the algebraic origin of integrability in lattice…
XXX one-magnon dispersion E(k) = 2J(1 - cos k); ratio E(π)/E(π/3) = 4
Exact one-magnon spectrum of the 1D XXX ferromagnet. Acting with H = -J·Σ S_i·S_{i+1} on the single-spin-flip state |k⟩ = (1/√N)·Σ_x…
KdV 1-soliton u = 2κ²·sech²(κ(x - 4κ²t)) satisfies u_t+6uu_x+u_xxx=0 exactly
Exact symbolic verification of the canonical Korteweg-de Vries one-soliton. Ansatz u(x, t) = 2κ²·sech²(κ(x - 4κ²·t)); differentiate u_t,…
6-vertex ice-rule anisotropy Δ = 1/2 (critical massless regime)
Closed-form evaluation of the Lieb 1967 6-vertex anisotropy Δ = (a²+b²-c²)/(2ab) at three canonical points. Ice point (Pauling 1935…