Algebraic and geometric properties of Bethe Ansatz eigenfunctions on a pentagonal magnetic ring

Magnetic Pentagonal Ring - Galois Extensions and Crystallography

“…for the magnetic pentagonal ring. Eigenstates and eigenvalues for this case were presented in [3] in terms of cyclotomic number field Q(ω), ω = exp(2π i /5), i.e. the extension of Q by the primitive fifth root of unity, associated with the Fourier transform for this case.…”

“…Still, Q(ω) is not sufficient to express this solution in the form prescribed by the Bethe Ansatz [4], that is in terms of spectral parameters of the Bethe pseudoparticles, or, equivalently, of the corresponding pseudomomenta or portions of phase. It was shown in [3] that the standard Bethe Ansatz presentation requires a further extension of the complex Heisenberg field to the so-called Bethe number field B, within the procedure referred there to as the inverse Bethe Ansatz: how to determine the Bethe parameters (pseudomomenta, etc.) once the exact solution of the Heisenberg eigenproblem is already known.…”

“…once the exact solution of the Heisenberg eigenproblem is already known. The inverse Bethe Ansatz for N = 5 was solved in [3], and the Galois symmetry associated with this case was thoroughly discussed in [5], in terms of a short exact sequence of automorphism group of consecutive field extensions, and appropriate homomorphism * corresponding author; e-mail: labuz@ur.edu.pl groups. This short exact sequence bears a strong analogy with crystallographic constructions of space groups as extensions of translation groups by point groups.…”

Point Group Interpretation of Galois Symmetry of Bethe Ansatz Solutions of Magnetic Pentagonal Ring

“…We intend to interpret them in terms of quantum mechanical notions and calculations. The main aim of the present paper is interpretation of the Galois group for the magnetic pentagon (N = 5) [5,6] in terms of some symmetries of point groups.…”

“…The eigenproblem of the pentagon and Bethe Ansatz A detailed description of the diagonalisation procedure for the magnetic pentagon has been given in [5]. Here we focus our attention on the two-magnon sector (r = 2 spin deviations from the ferromagnetic saturation), in the interior B int = {k = ±1, ±2} of the Brillouin zone for the pentagon, where k is the quasimomentum, the exact quantum number responsible for the translational symmetry of the pentagon, given by the cyclic group C 5 .…”