Singular solutions, repeated roots and completeness for higher-spin chains

Understand

We investigate the completeness of the solutions of the Bethe equations for the integrable spin-s isotropic (XXX) spin chain with periodic boundary conditions.

• Solutions containing the exact string i s, i (s-1), ..., -i(s-1), -is are singular.
• For s>1/2, there exist also "strange" solutions with repeated roots, which nevertheless are physical (i.e., correspond to eigenstates of the Hamiltonian).
• We derive conditions for the singular solutions and the solutions with repeated roots to be physical.