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The XXZ spin chain Bethe Ansatz equations are a complicated system of rational function equation:

\[ \left(\frac{\lambda_j + i/2}{\lambda_j - i/2} \right)^N = \prod_{l=1, l \neq j}^M \frac{\lambda_j - \lambda_l + i}{\lambda_j - \lambda_l - i} \]

Generically, since there are M equations and M unknowns generically we could say the solutions in $\lambda$ are a discrete set of points.

Is there any other logic to their roots besides the fact that they solve these equations ?

Spin chains have to do with the raising and lowering operators in the 2-dimensional representation of $SU(2)$ : \[\left.\begin{array}{cccc} S^+ : & |\uparrow \rangle & \mapsto & 0 \\ & |\downarrow \rangle & \mapsto & |\uparrow \rangle \\ \hline S^- : & |\uparrow \rangle & \mapsto & |\downarrow \rangle \\ & |\downarrow \rangle & \mapsto & 0 \\ \hline \\ S^z : & |\uparrow \rangle & \mapsto & \tfrac{1}{2}|\uparrow \rangle \\ & |\downarrow \rangle & \mapsto & \tfrac{1}{2}|\downarrow \rangle \end{array}\right. \]

The Hamiltonian acts on a tensor product of $SU(2)$ representations $V^{\otimes M}$.

\[ \mathcal{H} = - \frac{1}{2} \sum_{n=1}^M \bigg[ \sigma_n^x \sigma_{n+1}^x + \sigma_n^y \sigma_{n+1}^y + \Delta \sigma_n^z \sigma_{n+1}^z\bigg] \]

Various values of $\Delta$ have different interpretations. Since $[\mathcal{H}, S^z] = 0$ we can diagonalize in terms of the $\uparrow, \downarrow$ states. The Bethe Ansatz equations are the eigenvalue equations for $\mathcal{H}$ in this basis.

None of this tells you how to solve the eigenvalue equations, here they are.

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Sorry, S are sigmas ? Also Bethe eq. are not exactly for eigenvalues, eigenvalues can be expressed via them... – Alexander Chervov Jul 13 '13 at 15:47
If I read correctly $S^\pm = \sigma^x \pm i \sigma^y, S^z = \sigma^z$. The raising-lowering operators, being slightly different from Pauli spin matrices. – john mangual Jul 13 '13 at 16:00

The Bethe equations and Bethe ansatz are the central tools to study quantum integrable systems and there are thousands papers devoted to them. I cannot pretend to know substantial part of this research, so let me write some remarks which I am aware of.

You ask "... is there logic to their roots ... "

Yes, there is some logic - the keyword is "string hypothesis" (it is NOT related to string theory).

Let me quote from String hypothesis for gl(n|m) spin chains: a particle/hole democracy Section 3.1 page 10.

Suppose that N is large and some Bethe root $\lambda_n$ has a positive imaginary part. Then the l.h.s of (26) is exponentially large with N. To achieve this large value on the r.h.s. there should be another Bethe root $\lambda_n′ ∼ \lambda_n − i$, with the help of which the pole in the r.h.s. is created. Repeating the same arguments for $\lambda_n′$ and using the reality of solution of the Bethe Ansatz [36], we conclude that the Bethe roots are organized in the complexes of the type:

$\lambda_k = \lambda_0 + ik, ~~ k= -(s-1)/2, -(s-3)/2, ..., +(s-1)/2. $

where s is an integer. These complexes are called s-strings.

String hypothesis in its strong form states that all solutions of the Bethe Ansatz equations can be represented as a collection of strings, and that $\lambda_k$ are approximated by $\lambda_0 + ik$ values with exponential in N precision. In its strong form the string hypothesis is wrong. However there is an evidence that its weaker version is correct if the proper thermodynamic limit is taken. The weaker version states that most of the Bethe roots are organized into strings with exponential in N precision, and that the fraction of solutions which significantly differ from (27) decreases to 0 when N → ∞. We discuss in more details applicability of the string hypothesis in appendix A.

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