# Algebraic Bethe Ansatz state generator problem

Given $$B(\lambda)=T^0_1 (\lambda)$$ the component of the monodromy matrix T that creates a state, $$\lambda$$ the spectral parameter and $$| \Omega \rangle$$ the reference ground state,

In "Quantum Groups in two dimensional physics"(Gomez-Ruiz Altaba- Sierra), trying to write the Algebraic Bethe Ansatz State $$\Psi$$, it first says that if the reference state $$| \Omega \rangle$$ is formed by more than one spin state, I need to use the creation operator given by its coproduct, i.e. $$\Delta B(\lambda)$$ if I have two spins states. I then would say that if I have N spins states, the operator that I need use is $$\Delta^{(N-1)}B(\lambda)$$. Then it defines the Algebraic Bethe Ansatz State as: $$\Psi = \prod^M_{n=1} B(\lambda_n)|\Omega \rangle$$. I understand that the problem is that: 1. I need a set of spectral parameters $$\{ \lambda_n \}$$ 2. I need unique parameters. I don't understand:

1. Is it using the N-1 coproduct to create that state?
2. and if so, why is it not using a sum over n spin flipped rather than a product? A product doesn't make sense to me not even mathematically.
1. If it is NOT using the coproduct, when do I use the coproduct to obtain the right operator?? Thank you

## 1 Answer

The coproduct increases the number of sites $$N$$ (the length of the chain), while the product of different $$B$$s increases the number of excitations (magnons) $$M$$. So each $$B(\lambda_n)\equiv B^{(N)}(\lambda_n)=\Delta^{(N-1)} B(\lambda_n)$$ creates one excitation with quasimomentum $$p(\lambda_n)$$. The coproduct does not change the parameter $$\lambda_n$$.

Graphically: as in that book draw $$B$$ by a horizontal line (auxiliary space) to which we attach $$\lambda$$, with one vertical line (local physical space) for each $$N$$, so that each crossing is an $$R$$-matrix, and fix the values of the spins on the endpoints of the horizontal line to get $$B=T^0_1$$ out of the monodromy matrix $$T$$. Then the coproduct increases the number of vertical lines by one, while taking a product of such $$B$$s for given $$N$$ corresponds adding a horizontal line with a new spectral parameter. In other words, the product corresponds to putting diagrams of $$B$$s on top of each other, each of with has its own auxiliary spectral parameter $$\lambda_n$$, which represents the composition of operators, corresponding to taking their matrix products.