Paths vs. Winding Numbers for the Two-Magnon Sector of the XXX Heisenberg Magnet

Bethe Ansatz is the famous method of determination of eigenstates and eigenenergies for a wide range of quantum problems, e.g. for the Heisenberg XXX s=1/2 model. The Bethe equations applied to solve the problem of N nodes and r overturned spins on a magnetic chain are labeled by sets of winding numbers ${n_i}$, however the condition for admissible sets give an overcomplete number of results. On the other hand, combinatorial objects, so called "paths", give the exact number of eigenvectors for the problem described by (N,r) values. The paper presents the method of determining the set of winding numbers from the appropriate path for the sector of r=2 spin deviations