36 research outputs found
Towards a general solution to the linear Heisenberg problem
The construction of a complete set of stationary states of the linear Heisenberg system with periodic boundary conditions (rings) has been a topic of intensive research for many decades. Many eminent theoreticians have made their contribution to this joint eort, but we should mention in the rst place the epoch-making work of Bethe 1 and Hulthen. 5 Their work is the basis of many papers on this topic that have appeared during the past 70 years. We do no have the intention to give a complete survey of all this work but we will try to indicate that especially an asymptotic approach starting from the Hypothesis of Strings gives the prospect of a general and complete solution of the Heisenberg chain
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.Comment: 4 pages, RevTex. Latex2.09, amsfont
Hierarchical Dobinski-type relations via substitution and the moment problem
We consider the transformation properties of integer sequences arising from
the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form
exp(x (a*)^r a), r=1,2,..., under the composition of their exponential
generating functions (egf). They turn out to be of Sheffer-type. We demonstrate
that two key properties of these sequences remain preserved under
substitutional composition: (a)the property of being the solution of the
Stieltjes moment problem; and (b) the representation of these sequences through
infinite series (Dobinski-type relations). We present a number of examples of
such composition satisfying properties (a) and (b). We obtain new Dobinski-type
formulas and solve the associated moment problem for several hierarchically
defined combinatorial families of sequences.Comment: 14 pages, 31 reference
Magnetic translation groups in an n-dimensional torus
A charged particle in a uniform magnetic field in a two-dimensional torus has
a discrete noncommutative translation symmetry instead of a continuous
commutative translation symmetry. We study topology and symmetry of a particle
in a magnetic field in a torus of arbitrary dimensions. The magnetic
translation group (MTG) is defined as a group of translations that leave the
gauge field invariant. We show that the MTG on an n-dimensional torus is
isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x
Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible
unitary representations of the MTG on a three-torus and apply the
representation theory to three examples. We shortly describe a representation
theory for a general n-torus. The MTG on an n-torus can be regarded as a
generalization of the so-called noncommutative torus.Comment: 29 pages, LaTeX2e, title changed, re-organized, to be published in
Journal of Mathematical Physic
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 \omega^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.Comment: 13 pages, 7 figs, 5 tabs, revtex
Statistical Mechanics and the Physics of the Many-Particle Model Systems
The development of methods of quantum statistical mechanics is considered in
light of their applications to quantum solid-state theory. We discuss
fundamental problems of the physics of magnetic materials and the methods of
the quantum theory of magnetism, including the method of two-time temperature
Green's functions, which is widely used in various physical problems of
many-particle systems with interaction. Quantum cooperative effects and
quasiparticle dynamics in the basic microscopic models of quantum theory of
magnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the
spin-fermion model are considered in the framework of novel
self-consistent-field approximation. We present a comparative analysis of these
models; in particular, we compare their applicability for description of
complex magnetic materials. The concepts of broken symmetry, quantum
protectorate, and quasiaverages are analyzed in the context of quantum theory
of magnetism and theory of superconductivity. The notion of broken symmetry is
presented within the nonequilibrium statistical operator approach developed by
D.N. Zubarev. In the framework of the latter approach we discuss the derivation
of kinetic equations for a system in a thermal bath. Finally, the results of
investigation of the dynamic behavior of a particle in an environment, taking
into account dissipative effects, are presented.Comment: 77 pages, 1 figure, Refs.37
Mac Lane Method for Determination of Extensions of Finite Groups. I. A Review in the Context of Structure of Condensed Matter
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
Bethe Ansatz and Geometry of the Classical Configuration Space
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<F<r, point out all l-strings. The latter classify, in terms of rigged string configurations of Kerov, Kirillov and Reshetikhin, all exact Bethe eigenfunctions. In this way, we interpret these eigenfunctions in terms of the classical configuration space, in particular on the structure of islands of adjacent spin deviations, in a way independent of the size N