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1
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Mac Lane Method for Determination of Extensions of Finite Groups. I. A Review in the Context of Structure of Condensed Matter

100%
Lulek T.
Acta Physica Polonica A
|
1992
|
vol. 82
|
issue 3
377-393
EN
The problem of classification of all extensions of a finite group Q by an Abelian group T has been reviewed using cohomology of groups and fibre bundle picture. The relevance of this problem in condensed matter physics has been pointed out in the context of crystallography and gauge fields. The Mac Lane method of an effective construction of the corresponding second cohomology group as the quotient group of all operator homomorphisms vs. crossed homomorphisms from some free groups to T has been described in detail.
2
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Wigner-Racah Description of the Free-Electron Model

100%
Lulek T.
Acta Physica Polonica A
|
1991
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vol. 80
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issue 6
769-779
EN
A version of Wigner-Racah type of approach is proposed for the model of a free-electron gas in a cubic box. The approach bases on the structure of fibre bundle in description of states of a single electron. Contrary to the tight binding model, the base of the bundle is the reciprocal space rather than the positional one. An elementary quantity of the approach is the star of quasi-momentum. Such a treatment reveals the degeneracy of energy levels of the system of N electrons, in particular of the ground Fermi level, for an uncomplete filling of stars. The associated classification of energy levels and corresponding states can be performed within an atomic-like Wigner-Racah scheme.
3
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Spin-Orbit Coupling for f-Electrons in a Crystalline Field

100%
Lulek T.
|
EN
The notion of a spin-orbit interaction arises from consideration of dynamics of multielectron atoms, i.e. systems of N electrons in a spherical potential. This notion is essentially a single-particle one. We sketch its origin as a second-order correction when Dirac four-component wave equations for an electron in external electromagnetic fields are simplified to the two-component Pauli spinors. The constraints in spinorial degrees of freedom consist, roughly speaking, in neglecting the small component of the electron four-function. The spin-orbit interaction term serves to compensate effects of the small component. The crystalline field induces some deviations from spherical symmetry of an isolated atom, which yields some modifications of the spherical form of the spin-orbit interaction operator. These modifications can be described in terms of a number of tensor operators adapted to appropriate chains of subgroups of the spherical symmetry group. We present a classification of independent tensor operators and discuss the relevant parameters for f-ions.
4
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Wreath Product in Factorization of Holosymmetric Group

64%
Florek W., Lulek T.
Acta Physica Polonica A
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1991
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vol. 79
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issue 6
843-852
EN
The holosymmetric group Q of an n-dimensional crystal lattice determined by a given lattice basis B is considered. This group is contained in the n-dimensional orthogonal group O(n) so its elements preserve the orthogonality of basis vectors and their lengths. These conditions yield the decomposition of lattice basis into orthogonal sublattices and next the factorization of the holosymmetric group, which can be written as a direct product of complete monomial groups of k-dimensional (k ≤ n) holosymmetric groups. Simple, decomposable and primitive holosymmetric groups are discussed. The results for n ≤ 4 are presented.
5
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Mac Lane Method for Determination of Extensions of Finite Groups. II. An Example for the Dihedral Group D_{2}

64%
Lulek T., Chatterjee R.
Acta Physica Polonica A
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1992
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vol. 82
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issue 3
395-406
EN
The Mac Lane method of classification and construction of all extensions of a group Q by an Abelian group T is demonstrated on the case Q = D_{2}, T = C_{2}. Constructions involving free groups and operator homomorphisms are performed in detail, and the complete list of resulting extensions is given. It is shown that there are 8 classes of gauge equivalency, and they fall into 4 classes of isomorphism. The role of gauge transformations is pointed out. Physical contexts of various constructions are reviewed. A comparison with the direct cohomology definitions is performed.
6
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Coupled States of the System of Two Free Electrons on a Single Star

64%
Walcerz S., Lulek T.
Acta Physica Polonica A
|
1991
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vol. 80
|
issue 6
781-792
EN
A classification scheme of quantum states for the system of two free electrons in a cubic box, confined to a single star of quasi-momentum is proposed within a Racah-Wigner type of approach. Coupling of angular momenta of the atomic case is here substituted by Mackey theorem for transitive representations, which provides a crystalline analogue of orbital angular momentum - the resultant orbit of the geometric symmetry group. The action of the Pauli group is combined with that of the octahedral group which yields the connection between spin (i.e. the singlet or triplet pairing of electrons) and statistic of the positional factor. Resultant singlets and doublets - irreducible representations of the octahedral group, exhibit an ordinary Landau diamagnetic behaviour, whereas triplets are paramagnetic. A relation between the Mackey star and the star of resultant momentum is discussed.
7
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Magnetic Pentagonal Ring - Galois Extensions and Crystallography

64%
Labuz M., Lulek T.
Acta Physica Polonica A
|
2017
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vol. 132
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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.
8
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Racah Recoupling Matrices for System of Two Free Electrons Confined To Single Star

64%
Lulek T., Kłos J.
Acta Physica Polonica A
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1996
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vol. 90
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issue 3
483-490
EN
Racah-Wigner type of calculus is adapted to the system of two free electrons in a cubic box, confined to a single star p00 of quasimomentum. Various coupling schemes (LS-like, jj-like, and transitive) have been reviewed in analogy to the case of a free atom. Racah recoupling matrices are calculated explicitly, and their quantum-mechanical meaning is exposed in terms of octahedral multipoles.
9
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Rarefied Band in Spectrum of Two-Magnon States of Heisenberg Ferromagnet of Clock Dial Plate

64%
Lulek T., Wal A.
Acta Physica Polonica A
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1995
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vol. 88
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issue 2
259-268
EN
The Heisenberg isotropic exchange Hamiltonian with the nearest neighbour interactions for the ring of N = 12 nodes (the clock dial plate) with the spin 1/2 has been diagonalized within the manifold of all 66 two-magnon excitations from the ferromagnetic ground state. The spectrum consists of 5 full bands, i.e. bands defined over the whole Brillouin zone, and a single band which is doubly rarefied, i.e. defined only for even values of quasimomentum. The rarefied band is found to be dispersionless outside the centre of the Brillouin zone. It separates two bands of "weakly bounded" spin waves from two other which are "weakly antibounded". The last remaining band is situated below this collection of scattered spin waves, and describes two tightly bounded spin deviations. The spectrum exhibits dispersion, but it is degenerate (singlet and quintet) at the boundary of the Brillouin zone.
10
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Point Group Interpretation of Galois Symmetry of Bethe Ansatz Solutions of Magnetic Pentagonal Ring

51%
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.
11
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Some New Approaches of the XXX Heisenberg Model of the Two-Magnon Sector

51%
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.
12
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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
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2015
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vol. 128
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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.
13
Content available remote

Electronic Spaces and Classification of Multielectron States in Multicentre Systems

51%
Chatterjee R., Lulek T., Biel J.
Acta Physica Polonica A
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1992
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vol. 81
|
issue 4-5
467-474
EN
Electronic structure of a crystal within the tight binding model is described in terms of fibre bundle formalism, with the base and fiber being respectively the set of all sites and the single-centre space of electron spinorbitals. It is based on the Weyl's duality between the symmetric group and the unitary group, and paves the way for a Racah-Wigner type of description of electronic structure in multicentre systems.
14
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Bethe Ansatz and Geometry of the Classical Configuration Space

51%
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
15
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Galois Properties of the Eigenproblem of the Hexagonal Magnetic Heisenberg Ring

45%
Banaszak G., Barańczuk S., Lulek T., Milewski J., Stagraczyński R.
Acta Physica Polonica A
|
2012
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vol. 121
|
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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