A New Basis for Bethe Vectors of the Heisenberg Model

S. V. Semkin, V. P. Smagin

Evgeny Mukhin, V. Tarasov, A. Varchenko

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $°f = l, °g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.

Alexander N. Varchenko, Evgeny Mukhin, V. Tarasov

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional gl2-modules. As a byproduct we show that there exist exactly (nl) − (n l−1) two-dimensional vector subspaces V ⊂ C[u] with a basis f, g ∈ V such that deg f = l, deg g = n − l + 1 and f(u)g(u − 1) − f(u − 1)g(u) = (u + 1)n.

K. Isler, Manu B. Paranjape

Roger Haydock, C.M.M. Nex

Gwenaël Ferrando, Jules Lamers, Fedor Levkovich-Maslyuk +1 more

We study the trigonometric quantum spin-Calogero–Sutherland model, and the Haldane–Shastry spin chain as a special case, using a Bethe-Ansatz analysis. We harness the model’s Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero–Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe Ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero–Sutherland model that generalises the Yangian Gelfand–Tsetlin basis of Takemura–Uglov. The Bethe-Ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe Ansatz works in the presence of fusion, may be of independent interest.