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1
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Local Gauge and Magnetic Translation Groups

100%
Florek W.
Acta Physica Polonica A
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1997
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vol. 92
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issue 2
399-402
EN
The magnetic translation group was introduced as a set of operators T(R)=exp[-iR·(p-eA/c)/h]. However, these operators commute with the Hamiltonian for an electron in a periodic potential and a uniform magnetic field if the vector potential A (the gauge) is chosen in a symmetric way. It is showed that a local gauge field A_{R}(r) on a crystal lattice leads to operators, which commute with the Hamiltonian for any (global) gauge field A = A(r). Such choice of the local gauge determines a factor system ω(R,R') = T(R)T(R')T(R+R')^{-1}, which depends on a global gauge only. Moreover, for any potential A a commutator T(R)T(R')T(R)^{-1}T(R')^{-1} depends only on the magnetic field and not on the gauge.
2
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Product of Projective Representations in Description of Multi-Electron States in An External Magnetic Field

100%
Florek W.
|
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vol. 95
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issue 6
871-880
EN
In this paper all inequivalent irreducible projective representations of the two-dimensional translation group for a given factor system are determined. A normalized, i.e. corresponding to the Landau gauge, factor system is considered. Obtained representations directly lead to concept of magnetic cells and to periodicity with respect to the charge of a moving particle. It is also shown that the quantization condition is imposed on the product qH of the charge q and the magnitude of magnetic field H. The Kronecker product of such representations is considered and it is proven that the multiplication of representations corresponds to the addition of charges of particles moving in a given external magnetic field. In general, coupling of d representations corresponds to d-particle states. Presented results can be applied in any problem related to two-dimensional electron gas in a magnetic field, for example in the fractional quantum Hall effect or high temperature superconductivity.
3
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Irreducible Basis for Permutation Representations

100%
Florek W.
|
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vol. 96
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issue 6
699-712
EN
For a given finite group G its permutation representation P, i.e. an action on an n-element set, is considered. Introducing a vector space L as a set of formal linear combinations of | j 〉, 1 ≤ j ≤ n, the representation P is linearized. In general, the representation obtained is reducible, so it is decomposed into irreducible components. Decomposition of L into invariant subspaces is determined by a unitary transformation leading from the basis { | j 〉} to a new, symmetry adapted or irreducible, basis { |Γrγ〉}. This problem is quite generally solved by means of the so-called Sakata matrix. Some possible physical applications are indicated.
4
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Two-Dimensional Electron Gas in a Periodic Potential and External Magnetic Field: States of Pairs and Three-Particle Systems

100%
Florek W.
Acta Physica Polonica A
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2001
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vol. 99
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issue 5
601-610
EN
The group-theoretical classification of states of identical particle pairs is presented. Then obtained states are coupled with those of an antiparticle to construct states of a three-particle system. Investigations are performed using products of irreducible projective representations of the 2D translation group. For a given Born-von Kármán period N degeneracy of pair states is N, whereas three-particle states are N^2-fold degenerated. It has to be underlined that the case of even N is more complicated since pair states are labelled by four inequivalent irreducible projective representations. The problem of symmetry properties with respect to particles transposition is briefly discussed.
5
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Application of Algebraic Combinatorics to Finite Spin Systems

100%
Florek W.
Acta Physica Polonica A
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2001
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vol. 100
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issue 1
3-22
EN
A finite spin system invariant under a symmetry group G is a very illustrative example of a finite group action on mappings f: X → Y (X is a set of spin carriers, Y contains spin projections for a given spin number s). Orbits and stabilizers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not decrease a dimension of a given eigenproblem, but they label states in a systematic way. It allows construction of general formulas for vectors of symmetry adapted basis and matrix elements of operators commuting with the action of G in the space of states. The special role is played by double cosets, since they label nonequivalent (from the symmetry point of view) matrix elements ãx|H|yã for an operator H between Ising configurations |x〉,|y〉. Considerations presented in this paper should be followed by a detailed discussion of different symmetry groups (e.g.) cyclic or dihedral ones) and optimal implementation of algorithms. The paradigmatic example, i.e. a finite spin system, can be useful in investigations of magnetic macromolecules like Fe_6 or Mn_{12} acetate.
6
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Magnetic Translations for a Spatially Periodic Magnetic Field

100%
Florek W.
Acta Physica Polonica A
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2000
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vol. 97
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issue 4
619-628
EN
It is shown that in the case of a free electron in a spatially periodic magnetic field the concept of magnetic translations operators is still valid and, moreover, these operators can be defined in the same way as for a Bloch electron in a uniform magnetic field. The results can be a useful tool in the investigation of recently observed phenomena in 2D electron gas with spatially modulated density.
7
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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.
8
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Small Antiferromagnetic Spin Systems: Just beyond the Rotational Band Model

64%
Florek W., Kaliszan L.
Acta Physica Polonica A
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2015
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vol. 127
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issue 2
330-332
EN
Two systems of spins s=1/2 with the Heisenberg interactions are investigated: (i) an equilateral trapezoid and (ii) a regular hexagon. Both cases are compared with the corresponding sublattice Hamiltonians to determine splitting and mixing of energy levels with a given total spin of sublattices. It is shown that small modifications of the Hamiltonian parameters may significantly change (magnetic) properties of the eigenstates, especially probability of finding system in a state with determined value of the sublattice total spin.
9
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Equations of Motion for a Charged Particle in n-Dimensional Magnetic Field

64%
Florek W., Thomas M.
Acta Physica Polonica A
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2000
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vol. 97
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issue 5
935-938
EN
The equation of motion for a charged particle moving in the n-dimensional constant magnetic field is obtained for any linear gauge and any metric tensor by generalization of Johnson and Lipmann's approach. It allows to consider the magnetic orbits in the n-dimensional space. It is shown that the movement of a particle can always be decomposed into a number of two-dimensional cyclotronic motions and a free particle part.
10
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Universal Sequence of the Ground States and Energy Level Ordering in Frustrated Antiferromagnetic Rings with a Single Bond Defect

51%
Antkowiak M., Florek W., Kamieniarz G.
Acta Physica Polonica A
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2017
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vol. 131
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issue 4
890-892
EN
The universal sequence of the ground states for antiferromagnetic frustrated rings with the odd number of the local spins s and a single bond defect α described by the isotropic Heisenberg Hamiltonian is discussed. The Lieb-Mattis energy level ordering in a pentanuclear ring is revealed and the arising magnetisation steps are demonstrated.
11
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Application of Algebraic Combinatorics to Finite Spin Systems with Dihedral Symmetry

51%
Bucikiewicz S., Dębski L., Florek W.
Acta Physica Polonica A
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2001
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vol. 100
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issue 4
453-475
EN
Properties of a given symmetry group G are very important in investigation of a physical system invariant under its action. In the case of finite spin systems (magnetic rings as Fe_6, Cu_6, Fe_{10}, some planar macromolecules as Fe_{12} or Fe_8) the symmetry group is isomorphic with the dihedral group D_N. In this paper group-theoretical "parameters" of such groups are determined, especially decompositions of transitive representations into irreducible ones and double cosets. These results are necessary to construct matrix elements of any operator commuting with G in an efficient way. The approach proposed can be useful in many branches of physics, but here it is applied to finite spin systems, which serve as models for mesoscopic magnets.
12
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Highly Degenerated Ground States in Some Rings Modeled by the Ising Spins with Competing Interactions

51%
Florek W., Antkowiak M., Kamieniarz G., Jaśniewicz-Pacer K.
Acta Physica Polonica A
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2018
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vol. 133
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issue 3
411-413
EN
We discuss three Ising ring systems with competing interactions which are analogs of quantum systems and we show that they exhibit similar properties. In particular, the archetypal system of three antiferromagnetically coupled spins s has two magnetically degenerated ground states with |M|=s, when 0
13
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Collinear and Non-Collinear Configurations in Classical Geometrically Frustrated Ring-Shaped Systems

51%
Grajek M., Kopyciuk T., Florek W., Marlewski A.
Acta Physica Polonica A
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2018
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vol. 133
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issue 3
417-419
EN
Geometrically frustrated quantum spin systems, with competing antiferromagnetic couplings, show the Kahn degenerate frustration for some specific values of Heisenberg Hamiltonian parameters. It has been recently shown for rings with a defect bond and centered rings. In the case of classical counterparts of these systems, degenerated configurations with the lowest energy are present for the energy function parameter greater than a certain threshold. In these domains such configurations are planar but non-collinear with continuous changes of the net magnetic moment with respect to the Hamiltonian parameter. Outside these domains there is unique collinear ground state configuration (neglecting choice of the net magnetic moment direction). However, these collinear configurations are the same in both non-frustrated and geometrically frustrated domains. Numerically exact calculations for quantum systems strongly confirm that determined properties of their classical counterparts realize the classical limit s→∞.
14
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Modeling of the Experimental Molecular-Based Ring-Shaped Nanomagnets

39%
Antkowiak M., Kozłowski P., Musiał G., Florek W., Kamieniarz G., Esposito F.
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issue 5
965-966
EN
Quantum transfer matrix technique and numerically exact diagonalization method are applied to the Heisenberg spin systems to model ring-shaped molecules. Two cases are investigated: (i) a dozen of S = 1 spins with additional biquadratic exchange and (ii) a dimetallic molecule Cr_7Cd, where it is assumed that exchange anisotropy is determined in a local coordination system. In the latter case the calculated susceptibility is compared with experimental results.
15
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Magnet with Ferromagnetic Nearest-Neighbor and Antiferromagnetic Next-Nearest-Neighbor Exchange Interactions

33%
Szymczak R., Szymczak H., Kamieniarz G., Szukowski G., Jaśniewicz-Pacer K., Florek W., Maltsev V., Babonas G.
Acta Physica Polonica A
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2009
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vol. 115
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issue 5
925-930
EN
Magnetic properties of Pb[Cu(SO_4)(OH)_2] (linarite) natural single crystals were studied by magnetization and specific heat measurements. The angular dependences of magnetization were revealed which correlated with the regularities of the crystal structure. At about 2.8 K this quasi-one-dimensional Heisenberg system undergoes a phase transition to the long-range antiferromagnetic order with antiparallel magnetic moments aligned probably along the b-axis. The antiferromagnetic order is evidenced by the metamagnetic transition and pronounced λ-type anomaly at T_N in the specific heat. Using phenomenological modeling based on a quantum transfer-matrix method, we argue that at higher temperatures linarite is a quasi-one-dimensional system with competing ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchange interactions.
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