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…
Lax-pair spectral curve: L*psi = lambda*psi defines Riemann surface C_g with dim Omega^1(C_g) = g
Lax-pair spectral-curve framework (Lax 1968 Commun Pure Appl Math 21, 467; Novikov-Manakov-Pitaevskii-Zakharov Theory of Solitons 1984).…
sl(2) quadratic Casimir C_2 = j(j+1); U_q(sl(2)) Hopf algebra structure (R-matrix via comultiplication)
sl(2) quadratic Casimir + U_q(sl(2)) Hopf-algebra framework (Drinfeld 1985, Jimbo 1986). Setup: the Lie algebra sl(2) has generators H, E,…
Grassmannian Gr(2,4) Plücker relation: p_12*p_34 - p_13*p_24 + p_14*p_23 = 0 in P^5
Grassmannian-embedding Plücker-relation framework. Setup: the Grassmannian Gr(k, n) parametrizes k-dimensional subspaces of an…
Theorem: compact Riemann surface C_g has dim H^0(C_g, Omega^1) = g (algebraic-geometric invariant)
Theorem (Riemann-Roch genus-g holomorphic-differential canonical): on a compact Riemann surface C_g of genus g, the dimension of the space…
Theorem: sl(2) quadratic Casimir C_2 = j(j+1) evaluates to 3/4 at spin j = 1/2
Theorem (sl(2) spin-1/2 Casimir canonical): for spin-j irreducible representation of sl(2), the quadratic Casimir C_2 = H^2/2 + EF + FE…
Theorem: Gr(2,4) Plücker identity p_12*p_34 - p_13*p_24 + p_14*p_23 = 0 (algebraic identity in 2x4 matrix)
Theorem (Gr(2,4) Plücker algebraic-identity canonical): for any 2x4 matrix with rows a = (a_1, a_2, a_3, a_4), b = (b_1, b_2, b_3, b_4),…
GGKM (1967)
Gardner-Greene-Kruskal-Miura 1967 inverse-scattering for KdV; Lax 1968 Lax-pair; basis of soliton-equations modern theory.
KdV solitons (1965)
N Zabusky-M Kruskal 1965 'soliton'-coinage from KdV numerical experiments; modern integrable-PDE field; basis of FPU paradox resolution.
Bethe ansatz (1931)
H Bethe 1931 Heisenberg XXZ chain solution; modern integrable spin-chains + 2D vertex models; Yang-Baxter equation foundational.
Painleve property (1900)
P Painleve 1900 6 transcendents PI-PVI; modern integrability-test for ODE systems; appears in random-matrix Tracy-Widom + 2D Ising.
Yang-Baxter (1967)
C N Yang 1967 + R J Baxter 1972 master-equation for integrable models; modern algebraic-Bethe-ansatz + quantum-groups Drinfeld 1985.
Lax pair (1968)
P Lax 1968 dL/dt = [M, L] for KdV; isospectral-evolution; basis of inverse-scattering-transform; modern algebraic-geometry approach.
KdV soliton (Zabusky-Kruskal 1965)
N Zabusky-M Kruskal 1965 numerical-soliton-discovery; modern modern foundational text + inverse-scattering Gardner-Greene-Kruskal-Miura…
Lax pair (Lax 1968)
P Lax 1968 Lax-pair compatibility; modern modern foundational text + AKNS hierarchy + Hirota-1971 bilinearization + tau-functions.
Inverse scattering (GGKM 1967)
Gardner-Greene-Kruskal-Miura 1967 IST + Zakharov-Shabat 1972 NLSE; modern modern foundational text + AKNS + Riemann-Hilbert.
Yang-Baxter (1967)
C N Yang 1967 + Baxter 1972 Yang-Baxter-equation; modern modern foundational text + integrable-spin-chains + quantum-groups Drinfeld 1985.
Toda lattice (1967)
M Toda 1967 + Flaschka-Manakov 1974 Toda-Lax-pair; modern modern foundational text + non-abelian Toda + algebraic-geometry-Krichever 1977.
Painlevé (1900)
P Painlevé 1900 + Gambier 1910 6 equations; modern modern foundational text + isomonodromy + applications-random-matrix Tracy-Widom.