Jordanian deformation of the non-compact and $\mathfrak{sl}_2 $-invariant $XXX_{-1/2}$ spin-chain

Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and $\mathfrak{sl}_2$-invariant $XXX_{-1/2}$ spin-chain. Before the deformation, the seed model can be understood as a sector of the $\mathfrak{psu}(2,2|4)$-invariant spin-chain encoding the spectral problem of $\mathcal{N}=4$ super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact $U_q(\mathfrak{sl}_2)$-invariant $XXZ_{-1/2} $ spin-chain.