Galois Symmetries of Bethe Parameters for the Heisenberg Pentagon

Magnetic Pentagonal Ring - Galois Extensions and Crystallography

“…The analogy between Galois extensions associated with Bethe Ansatz eigenstates and crystallography bases on comparison of two exact sequences of groups and homomorphisms (we use here the notation of [5])…”

“…are mutually conjugated by inner automorphisms of the group GL(2, Z 2 ), and yield a non-trivial extension, namely the Galois group G of the Bethe number field, G = Aut (B/Q(ω)), |G| = 16, (10) described in detail in [5]. Each time, it is a semidirect product of the active and the passive group, with appropriate action.…”

“…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 eld C is algebraically complete, i.e., any polynomial with coecients in C has all its roots in C. However, in many cases, including our case of pentagon, this eld is redundant, in spite of the following facts: (a) the original 10 × 10 problem in the basis of magnetic congurations requires only the prime eld Q ⊂ C (since all matrix elements are integers), (b) the eective four 2 × 2 eigenproblems for the interior B int require the cyclotomic extension Q(ω) of the prime eld Q, (c) solutions of the eigenproblem are also expressible in Q(ω). Due to this fact, Q(ω) can be referred to as to the Heisenberg eld of magnetic pentagon [6,7].…”