Abstract
We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator Bgood(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.
Highlights
Symmetry which in particular allows one to obtain eigenvalues of the spin chain Hamiltonian in terms of the Bethe ansatz equations [1]
We provide an explicit expression for this operator as a polynomial in the monodromy matrix entries
In the SU(N ) Bethe ansatz there are several types of Bethe roots, in the examples we considered it is just the momentum-carrying Bethe roots which should be plugged into the Bgood operators in (1.1) to construct the states
Summary
As the results we present in this paper are based on the algebraic Bethe ansatz framework, we will describe its basic components for SU(N ) spin chains, and introduce relevant notation. A curious feature of the construction is that one can build eigenstates with the same operator Bgood but using a dual set of Bethe roots and acting on a different reference state. This property is directly related to the fact that one can use an arbitrary matrix K when building the operator Bgood via (3.10). In our construction there is no need to switch between the operators B and C, instead one can use the same operator Bgood and act on the different reference state using the dual set of Bethe roots,. This can be proven using the same arguments as for the construction with usual Bethe roots and a generic matrix K which we discissed above (see appendix A)
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