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We present an exact solution to the asymptotic Bethe equation of weaklyanisotropic Heisenberg spin chain, which is a set of non-linear algebraicequations. The solution describes the low-energy excitations aboveferromagnetic ground state with fixed magnetisation, and it has a closerelation to generalised Jacobi polynomial. It is equivalent to a generalisedStieltjes problem and in the continuous limit, it becomes a Riemann-Hilbertproblem closely related to the finite-gap solutions of classicalLandau-Lifshitz field theory.

An exact solution to asymptotic Bethe equation