Arithmetic of Partial Fibres in Relative Position Space of Bethe Ansatz

Milewski, Jan; Banaszak, Grzegorz; Lulek, T.

doi:10.1142/s1230161210000072

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…Notice that Lemma 4.1, although formulated in different context, is similar to Lemma 1 of [6]. Remark 4.10.…”

Section: Resultsmentioning

confidence: 96%

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Arithmetic of Heisenberg ring and cyclic group actions

Krasoń¹,

Milewski²

2019

Preprint

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“…Notice that Lemma 4.1, although formulated in different context, is similar to Lemma 1 of [6]. Remark 4.10.…”

Section: Resultsmentioning

confidence: 96%

“…In the paper [6] the number of elements in these fibers for some values of N were computed. It was shown, using certain natural group action, that for N, r coprime the fibers have the same number of elements.…”

Section: Introductionmentioning

confidence: 99%

Arithmetic of Heisenberg ring and cyclic group actions

Krasoń¹,

Milewski²

2019

Preprint

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“…Observe, that, according to simple combinatorial bijection, every position j in (3) can be treated as an r-element subset of ‹ N . Problems related to the classical Heisenberg magnetic configuration space with a fixed number of spin deviations were considered in papers [5]- [8]. Hence, the vector |Q, j can be written in the form |{j 1 , .…”

Section: R=0mentioning

confidence: 99%

“…, t r−1 ) -cf. [5]- [8]. We use the sequence which is lexically the first, and the integer 1 ≤ F ≤ r denotes the number of adjacent islands of Bethe pseudoparticles in the C 7 -orbit t [10].…”

Section: Galois Wavelets and Algebraic Structure Of The Cyclotomic Fieldmentioning

confidence: 99%

Galois Actions on the Eigenproblem of the Heisenberg Heptagon

Milewski

Banaszak

Lulek

et al. 2012

Open Syst. Inf. Dyn.

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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. Exemplary OSID style[J. Milewski et al., Galois actions on the eigenproblem of the Heisenberg heptagon] 2 Kummer theory.

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Galois Properties of the Eigenproblem of the Hexagonal Magnetic Heisenberg Ring

et al. 2012

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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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Arithmetic of Partial Fibres in Relative Position Space of Bethe Ansatz

Cited by 4 publications

References 8 publications

Arithmetic of Heisenberg ring and cyclic group actions

Arithmetic of Heisenberg ring and cyclic group actions

Galois Actions on the Eigenproblem of the Heisenberg Heptagon

Galois Properties of the Eigenproblem of the Hexagonal Magnetic Heisenberg Ring

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