Necklace Ansatz for strongly repulsive spin mixtures on a ring

We propose an alternative to the Bethe Ansatz method for strongly-interacting fermionic (or bosonic) mixtures on a ring. Starting from the knowledge of the solution for single-component non-interacting fermions (or strongly-interacting bosons), we explicitly impose periodic condition on the amplitudes of the spin configurations. This reduces drastically the number of independent complex amplitudes that we determine by constrained diagonalization of an effective Hamiltonian. This procedure allows us to obtain a complete basis for the exact low-energy many-body solutions for mixtures with a large number of particles, both for $SU(\kappa)$ and symmetry-breaking systems.