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Paths vs. Winding Numbers for the Two-Magnon Sector of the XXX Heisenberg Magnet

100%
Labuz M.
Acta Physica Polonica A
|
2015
|
vol. 128
|
issue 2
190-192
EN
Bethe Ansatz is the famous method of determination of eigenstates and eigenenergies for a wide range of quantum problems, e.g. for the Heisenberg XXX s=1/2 model. The Bethe equations applied to solve the problem of N nodes and r overturned spins on a magnetic chain are labeled by sets of winding numbers {n_i}, however the condition for admissible sets give an overcomplete number of results. On the other hand, combinatorial objects, so called "paths", give the exact number of eigenvectors for the problem described by (N,r) values. The paper presents the method of determining the set of winding numbers from the appropriate path for the sector of r=2 spin deviations.
2
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Magnetic Pentagonal Ring - Galois Extensions and Crystallography

63%
Labuz M., Lulek T.
Acta Physica Polonica A
|
2017
|
vol. 132
|
issue 1
97-99
EN
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.
3
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Matrix Representation of Constants of Motion for One-Dimensional Heisenberg Magnet

51%
Jakubczyk P., Labuz M., Lulek T.
Acta Physica Polonica A
|
2015
|
vol. 128
|
issue 2
176-178
EN
We demonstrate an exact diagonalization of the one-dimensional Heisenberg magnet in terms of algebraic Bethe Ansatz. We point out, by a polynomial expansion of the transfer matrix with respect to spectral parameter, a complete set of observables for classification of all eigenstates. We introduce an application of our approach on the example of the Heisenberg magnet consisting of four qubits, including its constants of motion, density matrices and complete classification of eigenstates.
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