275 research outputs found
An Exactly Solvable Spin Chain Related to Hahn Polynomials
We study a linear spin chain which was originally introduced by Shi et al.
[Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength
contains a parameter and depends on the parity of the chain site.
Extending the model by a second parameter , it is shown that the single
fermion eigenstates of the Hamiltonian can be computed in explicit form. The
components of these eigenvectors turn out to be Hahn polynomials with
parameters and . The construction of the
eigenvectors relies on two new difference equations for Hahn polynomials. The
explicit knowledge of the eigenstates leads to a closed form expression for the
correlation function of the spin chain. We also discuss some aspects of a
-extension of this model
A classification of generalized quantum statistics associated with basic classical Lie superalgebras
Generalized quantum statistics such as para-statistics is usually
characterized by certain triple relations. In the case of para-Fermi statistics
these relations can be associated with the orthogonal Lie algebra B_n=so(2n+1);
in the case of para-Bose statistics they are associated with the Lie
superalgebra B(0|n)=osp(1|2n). In a previous paper, a mathematical definition
of ``a generalized quantum statistics associated with a classical Lie algebra
G'' was given, and a complete classification was obtained. Here, we consider
the definition of ``a generalized quantum statistics associated with a basic
classical Lie superalgebra G''. Just as in the Lie algebra case, this
definition is closely related to a certain Z-grading of G. We give in this
paper a complete classification of all generalized quantum statistics
associated with the basic classical Lie superalgebras A(m|n), B(m|n), C(n) and
D(m|n)
A class of infinite-dimensional representations of the Lie superalgebra osp(2m+1|2n) and the parastatistics Fock space
An orthogonal basis of weight vectors for a class of infinite-dimensional
representations of the orthosymplectic Lie superalgebra osp(2m+1|2n) is
introduced. These representations are particular lowest weight representations
V(p), with a lowest weight of the form [-p/2,...,-p/2|p/2,...,p/2], p being a
positive integer. Explicit expressions for the transformation of the basis
under the action of algebra generators are found. Since the relations of
algebra generators correspond to the defining relations of m pairs of
parafermion operators and n pairs of paraboson operators with relative
parafermion relations, the parastatistics Fock space of order p is also
explicitly constructed. Furthermore, the representations V(p) are shown to have
interesting characters in terms of supersymmetric Schur functions, and a simple
character formula is also obtained.Comment: 15 page
Gel'fand-Zetlin basis for a class of representations of the Lie superalgebra gl(\infty|\infty)
A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible
covariant tensor representations of the Lie superalgebra gl(n|n). The related
Gel'fand-Zetlin patterns are based upon the decomposition according to a
particular chain of subalgebras of gl(n|n). This chain contains only genuine
Lie superalgebras of type gl(k|l) with k and l nonzero (apart from the final
element of the chain which is gl(1|0)=gl(1)). Explicit expressions for a set of
generators of the algebra on this Gel'fand-Zetlin basis are determined. The
results are extended to an explicit construction of a class of irreducible
highest weight modules of the general linear Lie superalgebra
gl(\infty|\infty).Comment: 21 page
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