arXiv subjects

Wojciech Florek

Publications and source records attributed to Wojciech Florek.

18 recordsLinked to original sources

arXiv2025

Ground states of classical spin polygons: Rigorous results and examples

We present a comprehensive and rigorous analysis of the lowest energy configurations (LECs) of classical spin polygons characterized by arbitrary couplings between neighboring spin sites. Our study shows that these ground states exhibit either collinear or coplanar arrangements, which allows us to determine the precise boundaries between these two phases. By simultaneously applying a spin flip and a bond inversion, we simplify the LEC problem and reduce it to a specific scenario with predominantly ferromagnetic (FM) bonds and a single antiferromagnetic (AFM) bond. Hence, competing interactions are always present, but, nevertheless, in the well-defined ranges of the system parameters the collinear LEC is realized. The difference angles between neighboring spins within the LEC can be captured by a single Lagrange parameter. We analytically investigate its dependence on the AFM bond and arrive at revealing results. Similarly, we can analyze the energy of the LEC, which shows a pronounced maximum as a function of AFM bond. To illustrate our findings, we give various examples that clearly demonstrate these results.

cond-mat.stat-mech↗

arXiv2020

Theory of ground states for classical Heisenberg spin systems VI

We formulate part VI of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. After recapitulating the central results of the parts I - V previously published we consider a magnetic field and analytically calculate the susceptibility at the saturation point. To this end we have to distinguish between parabolic and non-parabolic systems, and for the latter ones between two- and three-dimensional ground states. These results are checked for a couple of examples.

cond-mat.stat-mech↗

arXiv2020

Theory of ground states for classical Heisenberg spin systems V

We formulate part V of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. After recapitulating the central results of the parts I - IV previously published we extend the theory to the case where an involutary symmetry is present and the ground states can be distinguished according to their degree of mixing components of different parity. The theory is illustrated by a couple of examples of increasing complexity.

cond-mat.str-el↗

arXiv2019

Spectrum of some arrow-bordered circulant matrix

Given a circulant matrix $\mathrm{circ}(c,a,0,0,...,0,a)$, $a\ne 0$, of order~$n$, we ``border'' it from left and from above by constant column and row, respectively, and we set the left top entry to be $-nc$. This way we get a~particular title object, an example of what we call an \textit{abc matrix\/}, or an \textit{arrow-bordered circulant (matrix)\/}. We find its eigenpairs and we discuss its spectrum with stress on extreme eigenvalues and their bounds. At last we notice its relation to a~weighted wheel graph

math.CO↗

arXiv2002

Neel probability and spin correlations in some nonmagnetic and nondegenerate states of hexanuclear antiferromagnetic ring Fe6: Application of algebraic combinatorics to finite Heisenberg spin systems

The spin correlations ω^z_r, r=1,2,3, and the probability p_N$ of finding a system in the Neel state for the antiferromagnetic ring Fe(III)6 (the so-called `small ferric wheel') are calculated. States with magnetization M=0, total spin 0<=S<=15 and labeled by two (out of four) one-dimensional irreducible representations (irreps) of the point symmetry group D_6 are taken into account. This choice follows from importance of these irreps in analyzing low-lying states in each S-multiplet. Taking into account the Clebsch--Gordan coefficients for coupling total spins of sublattices (SA=SB=15/2) the global Neel probability p*_N can be determined. Dependencies of these quantities on state energy (per bond and in the units of exchange integral J) and the total spin S are analyzed. Providing we have determined p_N(S) etc. for other antiferromagnetic rings (Fe10, for instance) we could try to approximate results for the largest synthesized ferric wheel Fe18. Since thermodynamic properties of Fe6 have been investigated recently, in the present considerations they are not discussed, but only used to verify obtained values of eigenenergies. Numerical results re calculated with high precision using two main tools: (i) thorough analysis of symmetry properties including methods of algebraic combinatorics and (ii) multiple precision arithmetic library GMP. The system considered yields more than 45 thousands basic states (the so-called Ising configurations), but application of the method proposed reduces this problem to 20-dimensional eigenproblem for the ground state (S=0). The largest eigenproblem has to be solved for S=4; its dimension is 60. These two facts (high precision and small resultant eigenproblems) confirm efficiency and usefulness of such an approach, so it is briefly discussed here.

cond-mat.mes-hall↗

arXiv2002

Two-dimensional electron gas in a periodic potential and external magnetic field: states of pairs and three-particle systems

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 BvK 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.

cond-mat.mes-hall↗

arXiv2002

Center of the charged particle orbit for any linear gauge

In the case of a constant uniform magnetic field it can be assumed, without the loss of generality, that the vector potential (the gauge) is a linear function of position, i.e. it could be considered as a three-dimensional real matrix or, more generally in an n-dimensional space, as a tensor A of the rank two. The magnetic tensor H is obtained from A by antisymmetrization, i.e. H=A-A^T. It is shown that the transpose of A plays a special role, since it determines the operator of the orbit center of a charged particle moving in an external magnetic field H. Moreover, this movement can be considered as a combination of N<=n independent cyclotronic movements in orthogonal planes (cyclotron orbits) with quantized energies, whereas in other n-2N dimensions the particle is completely free with a continuous energy spectrum. The proposed approach enables introduction of the four-dimensional space-time and, after some generalizations, non-linear gauges.

quant-ph↗

arXiv2000

Trions in a periodic potential

The group-theoretical classification of trion states is presented. It is based on considerations of products of irreducible representations of the 2D translation group. For a given BvK period N degeneracy of obtained states is N^2. Trions consist of two identical particles so the symmetrization of states with respect to particles transposition is considered. Completely antisymmetric states can be constructed by introducing antisymmetric spin functions. Two symmetry adapted bases are considered. The third possibility is postponed for the further investigations.

cond-mat.mes-hall↗

arXiv1999

Magnetic translations for a spatially periodic magnetic field

It is shown that in the case of 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 investigation of lately observed phenomena in 2DEG with spatially modulated density.

cond-mat.mes-hall↗

arXiv1999

Algebraic description of a two-dimensional system of charged particles in an external magnetic field and periodic potential

Properties of the magnetic translation operators for a charged particle moving in a crystalline potential and a uniform magnetic field show that it is necessary to consider all inequivalent irreducible projective representations of the the crystal lattice translation group. These considerations lead to the concept of magnetic cells and indicate the periodicity of physical properties with respect to the charge. It is also proven that a direct product of such representations describe a system of two (many, in general) particles. Therefore, they can be applied in description of interacting electrons in a magnetic field, for example in the fractional quantum Hall effect.

cond-mat.mes-hall↗

arXiv1998

Magnetic translation groups as group extension

Extensions of a direct product T of two cyclic groups Z_n1 and Z_n2 by an Abelian (gauge) group G with the trivial action of T on G are considered. All possible (nonequivalent) factor systems are determined using the Mac Lane method. Some of resulting groups describe magnetic translation groups. As examples extensions with G=U(1) and G=Z_n are considered and discussed.

cond-mat.mes-hall↗

arXiv1998

Magnetic translation groups in n dimensions

Magnetic translation groups are considered as central extensions of the translation group T=Z^n by the group of factors (a~gauge group) U(1). The obtained general formulae allow to consider a magnetic field as an~antisymmetric tensor (of rank 2) and factor systems are determined by a transvection of this tensor with a tensor product t \otimes t'.

cond-mat.mes-hall↗

arXiv1998

Mac Lane method in the investigation of magnetic translation groups

Central extensions of the three-dimensional translation group T=Z^3 by the unitary group U(1) (a group of factors) are considered within the frame of the Mac~Lane method. All nonzero vectors t in T are considered to be generators of T. This choice leads to very illustrative relations between the Mac~Lane method and Zak's approach to magnetic translation groups. It is shown that factor systems introduced by Zak and Brown can be realized only for the unitary group U(1) and for some of its finite subgroups.

cond-mat.mes-hall↗

arXiv1998

Local gauge and magnetic translation groups

The magnetic translation group was introduced as a set of operators T(R)=\exp[-iR.(p-eA/c)/\hbar]. 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 afactor 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.

cond-mat.mes-hall↗

arXiv1998

Pairs of Bloch electrons and magnetic translation groups

A product of irreducible representations of magnetic translation group is considered. It leads to irreducible representations which were previously rejected as nonphysical. A very simple example indicates a possible application of these representations. In particular, they are important in descriptions of pairs of electrons in a magnetic field and a periodic potential. The periodicity of some properties with respect to the charge of a particle is briefly discussed.

cond-mat.mes-hall↗

arXiv1998

Combinatorics, wreath products, finite space groups and magnetism

In this lecture some mathematical tools necessary for a proper description of the Heisenberg antiferromagnet are presented. We would like to point out differences between ferro- and antiferromagnetic cases of Heisenberg Hamiltonian for finite spin systems. The ground-state properties are discussed.

math-ph↗

arXiv1998

The role of a form of vector potential - normalization of the antisymmetric gauge

Results obtained for the antisymmetric gauge A=[Hy,-Hx]/2 by Brown and Zak are compared with those based on pure group-theoretical considerations and corresponding to the Landau gauge A=[0,Hx]. Imposing the periodic boundary conditions one has to be very careful since the first gauge leads to a factor system which is not normalized. A period N introduced in Brown's and Zak's papers should be considered as a magnetic one, whereas the crystal period is in fact 2N. The `normalization' procedure proposed here shows the equivalence of Brown's, Zak's, and other approaches. It also indicates the importance of the concept of magnetic cells. Moreover, it is shown that factor systems (of projective representations and central extensions) are gauge-dependent, whereas a commutator of two magnetic translations is gauge-independent. This result indicates that a form of the vector potential (a gauge) is also important in physical investigations.

cond-mat.mes-hall↗

arXiv1997

Kronecker products of projective representations of translation groups

Projective irreps of (Z_N)^2 can be labelled by divisors n of N. A product of two irreps, labelled by n and n', can be decomposed into projective irreps labelled by M, where M strongly depends on the arithmetic structure of N, n, n' and their relations (gcd, lcm etc.). Such decompostion describes two important physical effects: (i) changes of a magnetic period of the crystal lattice (with unchaged crystal period N); (2) each representation can be related with a charged particle moving in an external magnetic field and a periodic potential --- a product of (projective) irreps corresponds to interaction of particles with charges Q and Q', respectively, and the decomposition corresponds to a particle with the charg Q''=Q+Q'.

cond-mat.mes-hall↗