The string hypothesis of the Bethe solutions of Heisenberg XXX model

I am studying L. Fadeev's "How Algebraic Bethe Ansatz works for integrable model". He takes Heisenberg $XXX_{1/2}$ model as an example. After obtaining the Bethe Ansatz Equations (BAE) for the roots {$\lambda$}, it is suggested the roots can be described by the string hypothesis in thermodynamic limit. The roots in a complex has the same real part $\lambda_{M,j}$, where $M,j$ means it is the $j$ th type-M complex.

Referring to Eq. 137 of the paper, there is an integer (half-integer) $Q_{M,j}$. $$2N\arctan(\frac{\lambda_{M,j}}{M+1/2})=2\pi Q_{M,j} + \sum_{M'}\sum_{(M',k)\neq(M,j)}\Phi_{M,M'}(\lambda_{M,j}-\lambda_{M',k})$$ and $$\Phi_{M,M'}(\lambda)=\sum_{L=|M-M'|}^{M+M'}\bigl(\arctan(\frac{\lambda}{L})+ \arctan(\frac{\lambda}{L+1})\bigl)$$ I think each $Q_{M,j}$ corresponds to each $\lambda_{M,j}$. Since there are different sets of {$\lambda_{M,j}$}, we can have different corresponding sets of {$Q_{M,j}$}. However, in the following part to discuss the bound of the $Q_{M,j}$, a quantity labelled by $Q_{M,\infty}$ was found by putting $\lambda_{M,j}\to\infty$ in Eq. 137. Then, it says " the maximal admissible $Q_{M,j}$ is given by $$Q_{M,max}= Q_{M,\infty}-(2M+1)$$ because complex of type $M$ has $2M+1$ roots." I cannot undersand how this statement is reached and also the actual meaning of $Q_{M,j}$.

I am very new to this area and therefore don't know my question is trivial or not. I will be very grateful if someone is willing to explain the meaning of $Q_{M,j}$ and the string hypothesis to me. I feel I am not understanding the things correctly. It is also my first time to ask a question here so I will be sorry if my question is too long or have any other problems.

The link of the paper is https://arxiv.org/pdf/hep-th/9605187.pdf. And the things I mentioned above are mainly in section 5 and 6.

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. – Qmechanic Jan 27 '18 at 8:34