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Journal of Physics Condensed Matter

Kinematics of the one-dimensional finite Heisenberg magnet with impurities

Barbara Lulek

Published Nov 9, 1992• 12 citations
Publisher

Abstract

The kinematics of the one-dimensional finite Heisenberg magnetic ring with impurities is discussed in the light of the general recipe of Weyl (1952). The cyclic group generating the ring, and the group of all its automorphisms, play the roles of the obvious and hidden symmetry respectively. The hidden symmetry group imposes a property of the distribution of quantum states of a translationally invariant ensemble of magnets over the finite Brillouin zone. Inhomogeneities of the distribution arise from irregular orbits of the action of the cyclic group on the set of all magnetic configurations of the ensemble. Size-dependent effects, like rarefied bands, are classified using the arithmetic structure of integers, (prime numbers, socles and arithmetic exponents), which is more appropriate than the linear order in the ring of integers.

DOI
10.1088/0953-8984/4/45/009

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The kinematics of the one-dimensional finite Heisenberg magnetic ring with impurities is discussed in the light of the general recipe of Weyl (1952). The cyclic group generating the ring, and the group of all its automorphisms, play the roles of the obvious and hidden symmetry respectively. The hidden symmetry group imposes a property of the distribution of quantum states of a translationally invariant ensemble of magnets over the finite Brillouin zone. Inhomogeneities of the distribution arise from irregular orbits of the action of the cyclic group on the set of all magnetic configurations of the ensemble. Size-dependent effects, like rarefied bands, are classified using the arithmetic structure of integers, (prime numbers, socles and arithmetic exponents), which is more appropriate than the linear order in the ring of integers.
Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.
Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view.For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups.This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics.Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N .The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra.The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.
A general formula for the number of linearly independent states from (2 s + 1)N-dimensional Hilbert space for a linear chain of N spins s corresponding to a wave number k has been derived. As shown, this density is inhomogeneous within the Brillouin zone, and its shape depends on the decomposition of N into prime factors. It proves that non all elementary excitations in a Heisenberg magnet can be adequately described in terms of quasiparticles, and in particular, they not necessarily exhibit a band structure.
Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.
We use the famous Weyl recipe to introduce Bethe Ansatz (BA) solutions for the two-magnon sector in the case of the Heisenberg magnet consisting of a small number N of nodes. In accordance with the Weyl recipe, the obvious symmetry group of the model is the translation group C N , whereas the group of all automorphisms Aut C N constitutes the hidden symmetry, and, in fact, manifests important symmetry properties of the system. With the use of the action of the Aut C N on the set of eigenvalues of BA, we propose the way of drawing energy bands for the problem being discussed.
We perform an exact diagonalization of the Heisenberg Hamiltonian for the magnetic ring consisting of N=6 nodes, compare the result with the Bethe Ansatz (BA) approach and verify the hypothesis of strings. We find and explain the failure of BA method in a single case of the highest weight eigenvector at the boundary of the Brillouin zone for the case of two spin deviations. The explanation involves the orbit structure of the corresponding set of magnetic configurations. We demonstrate that the finite size effects for N=6 only slightly deform the shape of asymptotic strings.