"Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models

• Main Authors: Faribault, Alexandre, Dimo, Claude
• Format: Journal Article
• Language: English
• Published: 16-12-2018
• Subjects: Mathematics - Mathematical Physics; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.1812.06428

2022 J. Phys. A: Math. Theor. 55 415205 In this work we construct the eigenstates of the most general spin-1/2
Richardson-Gaudin model integrable in an external magnetic field. This includes
the possibility for fully anisotropic XYZ coupling such that the $S^x_iS^x_j$,
$S^y_iS^y_j$ and $S^z_iS^z_j$ terms all have distinct coupling strengths. While
insuring that integrability is maintained in the presence of an external field
excludes the elliptic XYZ model which is only integrable at zero field, this
work still covers a wide class of fully anisotropic (XYZ) models associated
with non skew-symmetric r-matrices.
The eigenstates, as constructed here, do not require any usable Bethe ansatz
and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin
raising (Gaudin) operators have to be defined. Indeed, the eigenstates are
generically built only through the conserved charges which define the model of
interest and the specification of the set of eigenvalues defining the
particular eigenstate. Since these eigenvalues are, in general, solutions to a
simple set of quadratic equations, the proposed approach is simpler to
implement than any Bethe ansatz and, moreover, it remains completely identical
independently of the symmetries of the model. Indeed, the construction removes
any distinction between XYZ models and XXZ/XXX models and, generically, that
between models with or without U(1) so that any difficulties associated with
the use of a Bethe ansatz in any of these cases are avoided.