Wojciech Florek

Characterization of some mesoscopic rings: simulation techniques

Abstract The numerical quantum transfer-matrix method and an advanced algebraic and combinatorial approach applicable to the mesoscopic rings have been worked out. Characterization of the ring [Mn(hfac)2NITPh]6 with non-uniform spin is also presented.

Trions in a periodic potential

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 two-dimensional translation group. For a given BvK period N degeneracy of obtained states is N2. 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)/2 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 an 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 presented elsewhere.

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 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 describes a system of two (many, in general) particles. Therefore, they can be applied in a description of interacting electrons in a magnetic field, for example in the fractional quantum Hall effect.

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.

Non-perturbative Methods in Phenomenological Simulations of Ring-Shape Molecular Nanomagnets

Two non-perturbative numerically exact methods: exact diagonalization and quantum transfer matrix are applied to computationally complex Heisenberg-like spin models of ring shaped molecular nanomagnets and implemented in the high performance computing environment. These methods are applicable to the wide class of ring-shaped nanomagnets. For the hypothetical antiferromagnetic nanomagnet Ni\(_{12}\) the influence of single-ion anisotropy on the ground states is investigated. For Cr\(_8\) it is demonstrated that the alternation of the nearest-neighbor bilinear exchange couplings leads to small changes in the magnetic torque with respect to the uniformly coupled system. Specific heat and entropy for Cr\(_8\) are showed to be good indicators of crossing fields. The applicability of the Lande rule to both systems is checked.

Single-ion anisotropy in a four-spin system: mixing of S-states

A four-spin system with s=1 and the single-ion anisotropy, D∑j[sjz]2, is considered. When D≠0 the Hamiltonian of the system does not commute with S2 and, therefore, S cannot be used as an additional label of energy levels. In this work we concentrate on the problem of mixing states with different total spins S. The Hamiltonian matrix is transformed to the symmetry-adapted basis (with subspaces labeled by the irreducible representations of the symmetry group) and next, after solving the eigenproblem for S2, to the basis with vectors labeled by S. Each eigenproblem is solved exactly (at least numerically) and the eigenstates are expressed as ∑SaS¦S〉. The coefficients aS are analyzed, especially for their D-dependence. Even in such a small system different schemes of level mixing can be observed.

Spin correlations in the ground state of hexanuclear magnetic ring Fe6

Abstract 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 2 s +1 z -components − s ⩽ m ⩽ s 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 μ :{1,2,…, N }→{− s ,…, s −1, s }) 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= 5 2 . The ground state and some excited states (with the total spin S =0) are determined and spin correlations are calculated. The results obtained are compared with spin correlations in the Neel state | s ,− s , s ,− s , s ,− s 〉, the ground state of a classical antiferromagnetic system.