Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Physica Polonica A

Magnetic Pentagonal Ring - Galois Extensions and Crystallography

M. Łabuz, Tadeusz Lulek

Published Jul 1, 2017• 0 citations
Publisher

Abstract

The Galois symmetry of exact Bethe Ansatz eigenstates for magnetic pentagonal ring is shown to bear a close analogy to some crystallographic constructions.Automorphisms of number field extensions associated with these eigenstates prove to be related to choices of the Bravais cells in the finite crystal lattice Z2 × Z2, responsible for extension of the cyclotomic field by the Bethe parameters.

DOI
10.12693/aphyspola.132.97

Related papers

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

Barbara Lulek, Tadeusz Lulek, M. Łabuz +1 more

Exact solutions of the eigenproblem of the magnetic pentagonal ring exhibit the arithmetic symmetry expressed in terms of a Galois group of a nite extension of the prime eld Q of rationals.We propose here a geometric interpretation of this symmetry in the interior of the Brillouin zone, in terms of point groups.Explicitly, it is a subgroup of the direct product C4 × D4.We present also the appropriate irreducible representations of the group.

Acta Physica Polonica A•Feb 1, 2015•0 citations

Crystallographic interpretation of Galois symmetries for magnetic pentagonal ring

Jan Milewski, Tadeusz Lulek, M. Łabuz

Physica B Condensed Matter•Dec 11, 2016•3 citations

Heisenberg and Bethe Field Extensions Applied to Magnetic Rings

Grzegorz Banaszak, Dorota Blinkiewicz, Piotr Krasoń +1 more

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.

Acta Physica Polonica A•Mar 1, 2018•1 citation

Bethe Ansatz, Galois symmetries, and finite quantum systems

Tadeusz Lulek

We demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum $N$-body problem for the linear magnetic ring of $N$ spins $1/2$ within the XXX model. We point out the purely

Banach Center Publications•Jan 1, 2016•1 citation

Galois Actions on the Eigenproblem of the Heisenberg Heptagon

Jan Milewski, Grzegorz Banaszak, Tadeusz Lulek +2 more

We analyse the exact solution of the eigenproblem for the Heisenberg Hamiltonian of magnetic heptagon, i.e. the ring of N = 7 nodes, each with spin 1/2, within the XXX model with nearest neighbour interactions, from the point of view of finite extensions of the field ℚ of rationals. We point out, as the main result, that the associated arithmetic structure of these extensions makes natural an introduction of some Galois qubits. They are two-dimensional subspaces of the Hilbert space of the model, which admit a quantum informatic interpretation as elementary memory units for a (hypothetical) computer, based on their distinctive properties with respect to the action of related Galois group for indecomposable factors of the secular determinant. These Galois qubits are nested on the lattice of subfields which involves several minimal fields for determination of eigenstates (the complex Heisenberg field), spectrum (the real Heisenberg field), and Fourier transforms of magnetic configurations (the cyclotomic field, based on the simple 7th root of unity). The structure of the corresponding lattice of Galois groups is presented in terms of Kummer theory, and its physical interpretation is indicated in terms of appropriate permutations of eigenstates, energies, and density matrices.

arXiv:1406.5330•Open Systems & Information Dynamics•Jun 1, 2012•13 citations

Galois Symmetries of Bethe Parameters for the Heisenberg Pentagon

Grzegorz Banaszak, Barbara Lulek, Tadeusz Lulek +2 more

Reports on Mathematical Physics•Apr 1, 2013•10 citations
The Galois symmetry of exact Bethe Ansatz eigenstates for magnetic pentagonal ring is shown to bear a close analogy to some crystallographic constructions.Automorphisms of number field extensions associated with these eigenstates prove to be related to choices of the Bravais cells in the finite crystal lattice Z2 × Z2, responsible for extension of the cyclotomic field by the Bethe parameters.
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.
We demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum NNN-body problem for the linear magnetic ring of NNN spins 1/21/21/2 within the XXX model. We point out the purely
We analyse the exact solution of the eigenproblem for the Heisenberg Hamiltonian of magnetic heptagon, i.e. the ring of N = 7 nodes, each with spin 1/2, within the XXX model with nearest neighbour interactions, from the point of view of finite extensions of the field ℚ of rationals. We point out, as the main result, that the associated arithmetic structure of these extensions makes natural an introduction of some Galois qubits. They are two-dimensional subspaces of the Hilbert space of the model, which admit a quantum informatic interpretation as elementary memory units for a (hypothetical) computer, based on their distinctive properties with respect to the action of related Galois group for indecomposable factors of the secular determinant. These Galois qubits are nested on the lattice of subfields which involves several minimal fields for determination of eigenstates (the complex Heisenberg field), spectrum (the real Heisenberg field), and Fourier transforms of magnetic configurations (the cyclotomic field, based on the simple 7th root of unity). The structure of the corresponding lattice of Galois groups is presented in terms of Kummer theory, and its physical interpretation is indicated in terms of appropriate permutations of eigenstates, energies, and density matrices.