We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities
θℓ,
ℓ=1,…,L satisfy
θℓ+ℏ=θℓ′ for
ℓ<ℓ′. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of
θℓ. In particular, we consider the regime
θℓ+1/θℓ≫1 for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.
We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries).
Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities
θℓ,
ℓ=1,…,L satisfy
θℓ+ℏ=θℓ′ for
ℓ<ℓ′. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of
θℓ. In particular, we consider the regime
θℓ+1/θℓ≫1 for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.
Bethe Ansatz equations for spin-s Heisenberg spin chain with s≥1 are significantly more difficult to analyze than the spin-1/2 case, due to the presence of repeated roots. As a result, it is challenging to derive extra conditions for the Bethe roots to be physical and study the related completeness problem. In this paper, we propose the rational Q-system for the XXX _s s spin chain. Solutions of the proposed Q-system give all and only physical solutions of the Bethe Ansatz equations required by completeness. This is checked numerically and proved rigorously. The rational Q-system is equivalent to the requirement that the solution and the corresponding dual solution of the TQ-relation are both polynomials, which we prove rigorously. Based on this analysis, we propose the extra conditions for solutions of the XXX _s s Bethe Ansatz equations to be physical.