Crystallographic interpretation of Galois symmetries for magnetic pentagonal ring

Galois symmetry of exact Bethe Ansatz eigenstates for the magnetic pentagonal ring within the XXX model are investigated by a comparison with crystallographic constructions of space groups. It follows that the arithmetic symmetry of Bethe parameters for the interior of the Brillouin zone admits crystallographic interpretation, in terms of the periodic square Z2×Z2, that is the two-dimensional crystal lattice with Born-Karman period two in both directions.