The spin correlations ω z r , r = 1, 2, 3, and the probability pN of finding a system in the Néel... more The spin correlations ω z r , r = 1, 2, 3, and the probability pN of finding a system in the Néel 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 D6 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 Néel 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 pN(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 are 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.
The equation of motion for a charged particle moving in the ndimensional constant magnetic filed ... more The equation of motion for a charged particle moving in the ndimensional constant magnetic filed 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.
Irreducible projective representations of the translation group of a finite N × N two-dimensional... more Irreducible projective representations of the translation group of a finite N × N two-dimensional lattice can be labeled by symbols n, l; q , where N = νn, gcd(l, n) = 1 and q denotes an irreducible representation of Z 2 ν . Obtained matrices are n-dimensional and the factor system of this representation does not depend on q and equals m
In this lecture some mathematical tools necessary for a proper description of the Heisenberg anti... more 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.
It is shown that in the case of free electron in a spatially periodic magnetic field the concept ... more 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.
The group-theoretical classification of states of identical particle pairs is presented. Then obt... more 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.
The group-theoretical classification of multiparticle states(pairs of particles and charged excit... more The group-theoretical classification of multiparticle states(pairs of particles and charged excitons X^+-) is based on considerations of products of irreducible projective representations of the two-dimensional translation group. The states of a pair particle-antiparticle are non-degenerate, whereas, for a given Born-von Karman period N, degeneracy of pair states is N andthree-particle states are N^2-fold degenerated. The symmetrization of states with respect to particles transposition is considered. Three symmetry adapted bases for trions are considered: (i) the first is obtained from a direct conjugation of three representations; (ii) in the second approach the states of a electrically neutral pair particle-antiparticle are determined in the first step; (iii) the third possibility is to consider a pair of identical particles in the first step. In the discussion presented the Landau gauge A=[0,Hx,0] is used, but it is shown that the results obtained are gauge-independent. In additi...
Computational Methods in Science and Technology, 2001
Two combinatorial algorithms, generation of ordered partitions of N with no more than m parts and... more Two combinatorial algorithms, generation of ordered partitions of N with no more than m parts and decompositions of N-element set into subsets with cardinalities given by a partition [k] = [k 0 k 1 ... k m!1 ] are presented and their possible applications to finite (mesoscopic) spin systems are indicated. The flow charts, listings, and results of test runs are provided.
Computational Methods in Science and Technology, 2001
Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems ... more Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems of very large dimensions. Efficient, and as exact as possible, solutions of such eigenproblems are very difficult. A method leading to block diagonalization of Hamiltonian matrix is proposed in this paper. For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers etc.) are introduced and briefly discussed. Generation of these structures can be performed by means of algorithms being modifications of standard ones. Main ideas are presented in this paper, whereas the actual form of algorithms will be discussed elsewhere.
Properties of the magnetic translation operators for a charged particle moving in a crystalline p... more 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.
The group-theoretical classification of trion states (charged excitons X ± ) is presented. It is ... more The group-theoretical classification of trion states (charged excitons X ± ) is presented. It is based on considerations of products of irreducible projective representations of the 2D translation group. For a given BvK period N degeneracy of obtained states is N 2 . Trions X ± consist of two identical particles (holes or electrons), so the symmetrization of states with respect to particles transposition is considered. There are N (N + 1)/2 symmetric and N (N − 1) antisymmetric states. Completely antisymmetric states can be constructed by introducing antisymmetric and symmetric spin functions, respectively. Two symmetry adapted bases are considered: the first is obtained from a direct conjugation of three representations, whereas in the second approach the states of a electrically neutral pair hole-electron are determined in the first step. The third possibility, a conjugation of representations corresponding to identical particles in the first step, is postponed for the further investigations.
A finite spin system invariant under a symmetry group G is a very illustrative example of the fin... more A finite spin system invariant under a symmetry group G is a very illustrative example of the finite group action on a set of mappings f : X -Y : In the case of spin systems X is a set of spin carriers and Y contains 2s þ 1 z-components Àspmps for a given spin number s: Orbits and stabilisers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not lead to smaller eigenproblems, but they label states in a systematic way. This enables us to determine eigenstates of a spin operator with high (numerical) precision. Having eigenvectors expressed as linear combinations of the so-called Ising configurations (mappings m : f1; 2; y; Ng-fÀs; y; s À 1; sg) their properties can be investigated. In this paper such approach is applied to the small ''ferric wheel'' Fe 6 : a macromolecular antiferromagnet containing six iron (III) ions with the spin number s ¼ 5 2 : The ground state and some excited states (with the total spin S ¼ 0) are determined and spin correlations are calculated. The results obtained are compared with spin correlations in the N! eel state js; Às; s; Às; s; ÀsS; the ground state of a classical antiferromagnetic system. r
Finite spin models, applicable in investigation of mesoscopic rings, give rise to eigenproblems o... more Finite spin models, applicable in investigation of mesoscopic rings, give rise to eigenproblems of very large dimensions. Solutions of such eigenproblems, which are both accurate and efficient, are very difficult. A method, based on combinatorial and group-theoretical considerations, leading to block diagonalization of the Hamiltonian matrix is proposed in this paper. For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers, etc.) are introduced and briefly discussed. Generation of these structures can be performed by means of slightly modified standard algorithms. The main ideas of these modification are presented in this paper. Possible applications of multiple precision libraries to eigenproblems are also mentioned.
Properties of polynomial coefficients are applied to investigation of finite spin systems. Number... more Properties of polynomial coefficients are applied to investigation of finite spin systems. Number of spin-configurations states with a given total magnetization are calculated by recurrence formulas for polynomial coefficients and for sum of polynomial coefficients.
A relation between geometric and gauge symmetries for the Heisenberg model of magnet is pointed o... more A relation between geometric and gauge symmetries for the Heisenberg model of magnet is pointed out. The space of quantum states of the magnet exhibits the structure of a bundle, with the base consisting of nodes of the crystal, and the typical fiber spanned on the set of the single-node spin projections. The geometric symmetry group acts on the base, the gauge group -on the typical fiber, and all combined operations form the wreath product. In particular, all global gauge transformations yield the direct product group -a subgroup of the wreath product.
A finite spin system invariant under a symmetry group G is a very illustrative example of the fin... more A finite spin system invariant under a symmetry group G is a very illustrative example of the finite group action on a set of mappings f : X → Y . In the case of spin systems X is a set of spin carriers and Y contains 2s + 1 z-components −s ≤ m ≤ s for a given spin number s. Orbits and stabilizers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not lead to smaller eigenproblems, but they label states in a systematic way. Some combinatorial and group-theoretical structures, like double cosets and transitive representations, appear in a natural way. In such a system one can construct general formulas for vectors of symmetry adapted basis and matrix elements of operators commuting with the action of G in the space of states. Considerations presented in this paper should be followed by detailed discussion of different symmetry groups (e.g. cyclic of dihedral ones) and optimal implementation of algorithms. The paradigmatic example, i.e. a finite spin system, can be useful in investigation of magnetic macromolecules.
The spin correlations ω z r , r = 1, 2, 3, and the probability pN of finding a system in the Néel... more The spin correlations ω z r , r = 1, 2, 3, and the probability pN of finding a system in the Néel 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 D6 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 Néel 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 pN(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 are 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.
The equation of motion for a charged particle moving in the ndimensional constant magnetic filed ... more The equation of motion for a charged particle moving in the ndimensional constant magnetic filed 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.
Irreducible projective representations of the translation group of a finite N × N two-dimensional... more Irreducible projective representations of the translation group of a finite N × N two-dimensional lattice can be labeled by symbols n, l; q , where N = νn, gcd(l, n) = 1 and q denotes an irreducible representation of Z 2 ν . Obtained matrices are n-dimensional and the factor system of this representation does not depend on q and equals m
In this lecture some mathematical tools necessary for a proper description of the Heisenberg anti... more 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.
It is shown that in the case of free electron in a spatially periodic magnetic field the concept ... more 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.
The group-theoretical classification of states of identical particle pairs is presented. Then obt... more 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.
The group-theoretical classification of multiparticle states(pairs of particles and charged excit... more The group-theoretical classification of multiparticle states(pairs of particles and charged excitons X^+-) is based on considerations of products of irreducible projective representations of the two-dimensional translation group. The states of a pair particle-antiparticle are non-degenerate, whereas, for a given Born-von Karman period N, degeneracy of pair states is N andthree-particle states are N^2-fold degenerated. The symmetrization of states with respect to particles transposition is considered. Three symmetry adapted bases for trions are considered: (i) the first is obtained from a direct conjugation of three representations; (ii) in the second approach the states of a electrically neutral pair particle-antiparticle are determined in the first step; (iii) the third possibility is to consider a pair of identical particles in the first step. In the discussion presented the Landau gauge A=[0,Hx,0] is used, but it is shown that the results obtained are gauge-independent. In additi...
Computational Methods in Science and Technology, 2001
Two combinatorial algorithms, generation of ordered partitions of N with no more than m parts and... more Two combinatorial algorithms, generation of ordered partitions of N with no more than m parts and decompositions of N-element set into subsets with cardinalities given by a partition [k] = [k 0 k 1 ... k m!1 ] are presented and their possible applications to finite (mesoscopic) spin systems are indicated. The flow charts, listings, and results of test runs are provided.
Computational Methods in Science and Technology, 2001
Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems ... more Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems of very large dimensions. Efficient, and as exact as possible, solutions of such eigenproblems are very difficult. A method leading to block diagonalization of Hamiltonian matrix is proposed in this paper. For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers etc.) are introduced and briefly discussed. Generation of these structures can be performed by means of algorithms being modifications of standard ones. Main ideas are presented in this paper, whereas the actual form of algorithms will be discussed elsewhere.
Properties of the magnetic translation operators for a charged particle moving in a crystalline p... more 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.
The group-theoretical classification of trion states (charged excitons X ± ) is presented. It is ... more The group-theoretical classification of trion states (charged excitons X ± ) is presented. It is based on considerations of products of irreducible projective representations of the 2D translation group. For a given BvK period N degeneracy of obtained states is N 2 . Trions X ± consist of two identical particles (holes or electrons), so the symmetrization of states with respect to particles transposition is considered. There are N (N + 1)/2 symmetric and N (N − 1) antisymmetric states. Completely antisymmetric states can be constructed by introducing antisymmetric and symmetric spin functions, respectively. Two symmetry adapted bases are considered: the first is obtained from a direct conjugation of three representations, whereas in the second approach the states of a electrically neutral pair hole-electron are determined in the first step. The third possibility, a conjugation of representations corresponding to identical particles in the first step, is postponed for the further investigations.
A finite spin system invariant under a symmetry group G is a very illustrative example of the fin... more A finite spin system invariant under a symmetry group G is a very illustrative example of the finite group action on a set of mappings f : X -Y : In the case of spin systems X is a set of spin carriers and Y contains 2s þ 1 z-components Àspmps for a given spin number s: Orbits and stabilisers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not lead to smaller eigenproblems, but they label states in a systematic way. This enables us to determine eigenstates of a spin operator with high (numerical) precision. Having eigenvectors expressed as linear combinations of the so-called Ising configurations (mappings m : f1; 2; y; Ng-fÀs; y; s À 1; sg) their properties can be investigated. In this paper such approach is applied to the small ''ferric wheel'' Fe 6 : a macromolecular antiferromagnet containing six iron (III) ions with the spin number s ¼ 5 2 : The ground state and some excited states (with the total spin S ¼ 0) are determined and spin correlations are calculated. The results obtained are compared with spin correlations in the N! eel state js; Às; s; Às; s; ÀsS; the ground state of a classical antiferromagnetic system. r
Finite spin models, applicable in investigation of mesoscopic rings, give rise to eigenproblems o... more Finite spin models, applicable in investigation of mesoscopic rings, give rise to eigenproblems of very large dimensions. Solutions of such eigenproblems, which are both accurate and efficient, are very difficult. A method, based on combinatorial and group-theoretical considerations, leading to block diagonalization of the Hamiltonian matrix is proposed in this paper. For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers, etc.) are introduced and briefly discussed. Generation of these structures can be performed by means of slightly modified standard algorithms. The main ideas of these modification are presented in this paper. Possible applications of multiple precision libraries to eigenproblems are also mentioned.
Properties of polynomial coefficients are applied to investigation of finite spin systems. Number... more Properties of polynomial coefficients are applied to investigation of finite spin systems. Number of spin-configurations states with a given total magnetization are calculated by recurrence formulas for polynomial coefficients and for sum of polynomial coefficients.
A relation between geometric and gauge symmetries for the Heisenberg model of magnet is pointed o... more A relation between geometric and gauge symmetries for the Heisenberg model of magnet is pointed out. The space of quantum states of the magnet exhibits the structure of a bundle, with the base consisting of nodes of the crystal, and the typical fiber spanned on the set of the single-node spin projections. The geometric symmetry group acts on the base, the gauge group -on the typical fiber, and all combined operations form the wreath product. In particular, all global gauge transformations yield the direct product group -a subgroup of the wreath product.
A finite spin system invariant under a symmetry group G is a very illustrative example of the fin... more A finite spin system invariant under a symmetry group G is a very illustrative example of the finite group action on a set of mappings f : X → Y . In the case of spin systems X is a set of spin carriers and Y contains 2s + 1 z-components −s ≤ m ≤ s for a given spin number s. Orbits and stabilizers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not lead to smaller eigenproblems, but they label states in a systematic way. Some combinatorial and group-theoretical structures, like double cosets and transitive representations, appear in a natural way. In such a system one can construct general formulas for vectors of symmetry adapted basis and matrix elements of operators commuting with the action of G in the space of states. Considerations presented in this paper should be followed by detailed discussion of different symmetry groups (e.g. cyclic of dihedral ones) and optimal implementation of algorithms. The paradigmatic example, i.e. a finite spin system, can be useful in investigation of magnetic macromolecules.
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