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Acta Physica Polonica A

Galois Properties of the Eigenproblem of the Hexagonal Magnetic Heisenberg Ring

Grzegorz Banaszak, Stefan Barańczuk, Tadeusz Lulek, Jan Milewski, Ryszard Stagraczyński

Published May 1, 2012• 7 citations
Publisher

Abstract

We analyse the number eld-theoretic properties of solutions of the eigenproblem of the Heisenberg Hamiltonian for the magnetic hexagon with the single-node spin 1/2 and isotropic exchange interactions.It follows that eigenenergies and eigenstates are expressible within an extension of the prime eld Q of rationals of degree 2 3 and 2 4 , respectively.In quantum information setting, each real extension of rank 2 represents an arithmetic qubit.We demonstrate in detail some actions of the Galois group on the eigenproblem.

DOI
10.12693/aphyspola.121.1111

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We analyse the number eld-theoretic properties of solutions of the eigenproblem of the Heisenberg Hamiltonian for the magnetic hexagon with the single-node spin 1/2 and isotropic exchange interactions.It follows that eigenenergies and eigenstates are expressible within an extension of the prime eld Q of rationals of degree 2 3 and 2 4 , respectively.In quantum information setting, each real extension of rank 2 represents an arithmetic qubit.We demonstrate in detail some actions of the Galois group on the eigenproblem.
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We consider striking connections between the theory of homogenous isotropic Heisenberg ring (XXX-model) and algebraic number theory.We explain the nature of these connections especially applications of Galois theory for computation of the spectrum of the Heisenberg operators and Bethe parameters.The solutions of the Heisenberg eigenproblem and Bethe Ansatz generate interesting families of algebraic number fields.Galois theory yields additional symmetries which not only simplify the analysis of the model but may lead to new applications and horizons.
Employing the method of a symmetry lowering (form the permutation group Pn to the dihedral group Cnv), the eigenstates of the ring Heisenberg Hamiltonian H=-2J ∑ni=1Si·Si+1 are classified not only by total spin S and its z-component M, but also by wave number k=2 πL/n, where L= ±0, ±1, ±2, ∼, ±p for n=2p+1 or 2p. By the characters of irreducible representation of Pn and Cnv, we express the sum of the energy eigenvalues at fixed S, M and L, together with the number of the eigenstates contained in this sum. With respect to the spin configurations in the ring, it is revealed that there exist invariant subspaces under all symmetry operations of Cnv in the space of the spin-functions specified by M. By connecting this fact with the Wigner coefficients, the eigenfunctions are easily obtained.
The kinematics of the one-dimensional finite Heisenberg magnetic ring with impurities is discussed in the light of the general recipe of Weyl (1952). The cyclic group generating the ring, and the group of all its automorphisms, play the roles of the obvious and hidden symmetry respectively. The hidden symmetry group imposes a property of the distribution of quantum states of a translationally invariant ensemble of magnets over the finite Brillouin zone. Inhomogeneities of the distribution arise from irregular orbits of the action of the cyclic group on the set of all magnetic configurations of the ensemble. Size-dependent effects, like rarefied bands, are classified using the arithmetic structure of integers, (prime numbers, socles and arithmetic exponents), which is more appropriate than the linear order in the ring of integers.