A combinatoric description of Bethe Ansatz solutions for nanoscopic systems

Bethe Ansatz provides an exact classification of eigenstates of the Heisenberg Hamiltonian for a finite magnetic ring, consisting of N nodes, each with the spin s = 1/2 (with some extensions to an arbitrary spin s) in terms of rigged string configurations. The latter are some combinatorial objects which serve as classification labels for solutions of Bethe equations. An astonishing feature is existence of Robinson-Schensted (RS) and Kerov-Kirillov-Reshetikhin (KKR) bijections between sets of (i) all magnetic configurations, (ii) all pairs of standard Young and Weyl tableaux of N boxes and n = 2s + 1 rows, (iii) all rigged string configurations, for given N and s. These bijections allow to point out an exact correspondence between physically admissible solutions of highly nonlinear Bethe Ansatz equations and the initial basis of quantum calculations - magnetic configurations which are just possible distributions of spins over the nodes.