72,820 research outputs found
Nematic phases and the breaking of double symmetries
In this paper we present a phase classification of (effectively)
two-dimensional non-Abelian nematics, obtained using the Hopf symmetry breaking
formalism. In this formalism one exploits the underlying double symmetry which
treats both ordinary and topological modes on equal footing, i.e. as
representations of a single (non-Abelian) Hopf symmetry. The method that exists
in the literature (and is developed in a paper published in parallel) allows
for a full classification of defect mediated as well as ordinary symmetry
breaking patterns and a description of the resulting confinement and/or
liberation phenomena. After a summary of the formalism, we determine the double
symmetries for tetrahedral, octahedral and icosahedral nematics and their
representations. Subsequently the breaking patterns which follow from the
formation of admissible defect condensates are analyzed systematically. This
leads to a host of new (quantum and classical) nematic phases. Our result
consists of a listing of condensates, with the corresponding intermediate
residual symmetry algebra and the symmetry algebra characterizing the effective
``low energy'' theory of surviving unconfined and liberated degrees of freedom
in the broken phase. The results suggest that the formalism is applicable to a
wide variety of two dimensional quantum fluids, crystals and liquid crystals.Comment: 17 pages, 2 figures, correction to table VII, updated reference
Weight bases of Gelfand-Tsetlin type for representations of classical Lie algebras
This paper completes a series devoted to explicit constructions of
finite-dimensional irreducible representations of the classical Lie algebras.
Here the case of odd orthogonal Lie algebras (of type B) is considered (two
previous papers dealt with C and D types). A weight basis for each
representation of the Lie algebra o(2n+1) is constructed. The basis vectors are
parametrized by Gelfand--Tsetlin-type patterns. Explicit formulas for the
matrix elements of generators of o(2n+1) in this basis are given. The
construction is based on the representation theory of the Yangians. A similar
approach is applied to the A type case where the well-known formulas due to
Gelfand and Tsetlin are reproduced.Comment: 29 pages, Late
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