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On Generalized Landau Levels

100%
Krasoń P., Milewski J.
Acta Physica Polonica A
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2017
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vol. 132
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issue 1
94-96
EN
We consider the dispersion of energy levels for both standard and inverted quantum harmonic oscillators in the presence of a uniform electromagnetic field. For this analysis we use a solution of the corresponding eigenproblem in terms of the Kummer functions. We find a complete description of the energy levels for a particle of mass m and electric charge q subject to the action of a harmonic oscillator and simultaneous uniform magnetic and electric fields. We also analyze the effect of spin on energy levels for an electron.
2
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Point Group Interpretation of Galois Symmetry of Bethe Ansatz Solutions of Magnetic Pentagonal Ring

81%
Lulek B., Lulek T., Łabuz M., Milewski J.
Acta Physica Polonica A
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2015
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vol. 127
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issue 2
336-338
EN
Exact solutions of the eigenproblem of the magnetic pentagonal ring exhibit the arithmetic symmetry expressed in terms of a Galois group of a finite extension of the prime field 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.
3
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Singularities of Bethe Ansatz via Robinson Numbers

81%
Koper A., Krasoń P., Milewski J.
Acta Physica Polonica A
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2018
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vol. 133
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issue 3
438-440
EN
In this work we suggest a rigorous mathematical approach for explanation of singular solutions of Bethe Ansatz by means of Robinson complex hypernumbers. There are several approaches towards these singular solutions eg. formal infinitesimals or germs of meromorphic functions. Our aim is to make them precise using non-standard analysis and show that they are essentially equivalent.
4
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Heisenberg and Bethe Field Extensions Applied to Magnetic Rings

81%
Banaszak G., Blinkiewicz D., Krasoń P., Milewski J.
Acta Physica Polonica A
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2018
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vol. 133
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issue 3
441-443
EN
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.
5
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Some New Approaches of the XXX Heisenberg Model of the Two-Magnon Sector

81%
Łabuz M., Milewski J., Lulek T.
Acta Physica Polonica A
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2018
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vol. 133
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issue 3
444-446
EN
XXX Heisenberg s-1/2 model has been examined in detail during last decades, however, recently one may find some new insights into that issue. Among several approaches describing the eigenproblem for the finite case, a close look into the structure of Bethe equations (BE) for the two-magnon sector case seems to be particularly interesting. BE enable us to evaluate parameters labeling eigenstates of a magnet, however to find appropriate sets of winding numbers, which parametrize BE, one has to apply the Inverse Bethe Ansatz method. On the other hand, one may choose a different - combinatoric approach - which also parametrizes Bethe eigenstates, with the use of rigging numbers describing string configurations. We present an idea of comparison of the concepts mentioned above for the particular case of two-spin deviations sector.
6
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Bethe Ansatz and Geometry of the Classical Configuration Space

81%
Lulek B., Lulek T., Milewski J.
Acta Physica Polonica A
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2009
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vol. 115
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issue 1
159-161
EN
We demonstrate that the seminal one-dimensional model of the Heisenberg magnet, consisting of N spins 1/2 with the nearest-neighbour isotropic interaction, solved exactly by Bethe ansatz, admits an interpretation of a system of r=N/2-M pseudoparticles (spin deviations) which are indistinguishable, have hard cores and move on the chain by local hoppings. Such an approach allows us to construct a manifold with some boundaries, which is genericly r-dimensional, and whose F-dimensional regions, 0
7
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Galois Properties of the Eigenproblem of the Hexagonal Magnetic Heisenberg Ring

71%
Banaszak G., Barańczuk S., Lulek T., Milewski J., Stagraczyński R.
Acta Physica Polonica A
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2012
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vol. 121
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issue 5-6
1111-1114
EN
We analyse the number field-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 field 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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