© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1406.5330 · Open Systems & Information Dynamics

Galois Actions on the Eigenproblem of the Heisenberg Heptagon

Jan Milewski, Grzegorz Banaszak, Tadeusz Lulek, Mirosław Łabuz, Ryszard Stagraczyński

Published Jun 1, 2012• 13 citations

Abstract

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.

DOI: 10.1142/s1230161212500126

Related papers

Galois Properties of the Eigenproblem of the Hexagonal Magnetic Heisenberg Ring

Grzegorz Banaszak, Stefan Barańczuk, Tadeusz Lulek +2 more

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.

Acta Physica Polonica A•May 1, 2012•7 citations

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

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

Magnetic Pentagonal Ring - Galois Extensions and Crystallography

M. Łabuz, Tadeusz Lulek

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.

Acta Physica Polonica A•Jul 1, 2017•0 citations

Moment Maps and Galois Orbits in Quantum Information Theory

Kael Dixon, Simon M Salamon

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb{C}^d$. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm{U}(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

arXiv:1912.03209•SIAM Journal on Applied Algebra and Geometry•Jan 1, 2020•1 citation

Separation of spherically and translationally covariant finite quantum spaces within the XXX model

Tadeusz Lulek, Ryszard Stagraczyński, M. Łabuz

A method of separation of subspaces with definite values of the total spin, its z-projection and quasimomentum, within the state space of the XXX model, i.e. the N-th tensor power of a single qubit, is presented. We construct a complete system of orthogonal projectors onto such subspaces by use of the basis of wavelets and chosen measuring operators based on the total spin. We also demonstrate, on examples of N=6 and 8 magnetic spin rings, that the constructed projectors can be further decomposed into the orthogonal sum of density matrices of eigenstates of the XXX model in the corresponding subspaces. This decomposition exhibits a Galois symmetry, stemming from roots of indecomposable factors of characteristic polynomial of the Heisenberg Hamiltonian. In this way, we propose an alternative for Bethe Ansatz for moderate lengths of spin chains, by a return to the original spectral problem, avoiding the highly nonlinear system of Bethe Ansatz equations. Such results are easy to adapt in quantum information processing.

Nuclear Physics B•Apr 28, 2023•1 citation

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.

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.

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.

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on

\mathbb{C}^d

. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in

\mathrm{U}(d)

that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

A method of separation of subspaces with definite values of the total spin, its z-projection and quasimomentum, within the state space of the XXX model, i.e. the N-th tensor power of a single qubit, is presented. We construct a complete system of orthogonal projectors onto such subspaces by use of the basis of wavelets and chosen measuring operators based on the total spin. We also demonstrate, on examples of N=6 and 8 magnetic spin rings, that the constructed projectors can be further decomposed into the orthogonal sum of density matrices of eigenstates of the XXX model in the corresponding subspaces. This decomposition exhibits a Galois symmetry, stemming from roots of indecomposable factors of characteristic polynomial of the Heisenberg Hamiltonian. In this way, we propose an alternative for Bethe Ansatz for moderate lengths of spin chains, by a return to the original spectral problem, avoiding the highly nonlinear system of Bethe Ansatz equations. Such results are easy to adapt in quantum information processing.