395 research outputs found
Representations of the quantum doubles of finite group algebras and solutions of the Yang--Baxter equation
Quantum doubles of finite group algebras form a class of quasi-triangular
Hopf algebras which algebraically solve the Yang--Baxter equation. Each
representation of the quantum double then gives a matrix solution of the
Yang--Baxter equation. Such solutions do not depend on a spectral parameter,
and to date there has been little investigation into extending these solutions
such that they do depend on a spectral parameter. Here we first explicitly
construct the matrix elements of the generators for all irreducible
representations of quantum doubles of the dihedral groups . These results
may be used to determine constant solutions of the Yang--Baxter equation. We
then discuss Baxterisation ans\"atze to obtain solutions of the Yang--Baxter
equation with spectral parameter and give several examples, including a new
21-vertex model. We also describe this approach in terms of minimal-dimensional
representations of the quantum doubles of the alternating group and the
symmetric group .Comment: 19 pages, no figures, changed introduction, added reference
Solutions of the Yang-Baxter equation: descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras
The representation theory of the Drinfeld doubles of dihedral groups is used
to solve the Yang-Baxter equation. Use of the 2-dimensional representations
recovers the six-vertex model solution. Solutions in arbitrary dimensions,
which are viewed as descendants of the six-vertex model case, are then obtained
using tensor product graph methods which were originally formulated for quantum
algebras. Connections with the Fateev-Zamolodchikov model are discussed.Comment: 34 pages, 2 figure
Bethe ansatz solution of an integrable, non-Abelian anyon chain with D(D_3) symmetry
The exact solution for the energy spectrum of a one-dimensional Hamiltonian
with local two-site interactions and periodic boundary conditions is
determined. The two-site Hamiltonians commute with the symmetry algebra given
by the Drinfeld double D(D_3) of the dihedral group D_3. As such the model
describes local interactions between non-Abelian anyons, with fusion rules
given by the tensor product decompositions of the irreducible representations
of D(D_3). The Bethe ansatz equations which characterise the exact solution are
found through the use of functional relations satisfied by a set of mutually
commuting transfer matrices.Comment: 19 page
Quantum Phase Transition in Finite-Size Lipkin-Meshkov-Glick Model
Lipkin model of arbitrary particle-number N is studied in terms of exact
differential-operator representation of spin-operators from which we obtain the
low-lying energy spectrum with the instanton method of quantum tunneling. Our
new observation is that the well known quantum phase transition can also occur
in the finite-N model only if N is an odd-number. We furthermore demonstrate a
new type of quantum phase transition characterized by level-crossing which is
induced by the geometric phase interference and is marvelously periodic with
respect to the coupling parameter. Finally the conventional quantum phase
transition is understood intuitively from the tunneling formulation in the
thermodynamic limit.Comment: 4 figure
SU(3) Richardson-Gaudin models: three level systems
We present the exact solution of the Richardson-Gaudin models associated with
the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for
any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For
the rational case additional cubic integrals of motion are obtained, whose
number is added to that of the quadratic ones to match, as required from the
integrability condition, the number of quantum degrees of freedom of the model.
We discuss different SU(3) physical representations and elucidate the meaning
of the parameters entering in the formalism. By considering a bosonic mapping
limit of one of the SU(3) copies, we derive new integrable models for three
level systems interacting with two bosons. These models include a generalized
Tavis-Cummings model for three level atoms interacting with two modes of the
quantized electric field.Comment: Revised version. To appear in Jour. Phys. A: Math. and Theo
A variational approach for the Quantum Inverse Scattering Method
We introduce a variational approach for the Quantum Inverse Scattering Method
to exactly solve a class of Hamiltonians via Bethe ansatz methods. We undertake
this in a manner which does not rely on any prior knowledge of integrability
through the existence of a set of conserved operators. The procedure is
conducted in the framework of Hamiltonians describing the crossover between the
low-temperature phenomena of superconductivity, in the
Bardeen-Cooper-Schrieffer (BCS) theory, and Bose-Einstein condensation (BEC).
The Hamiltonians considered describe systems with interacting Cooper pairs and
a bosonic degree of freedom. We obtain general exact solvability requirements
which include seven subcases which have previously appeared in the literature.Comment: 18 pages, no eps figure
Ground-state properties of the attractive one-dimensional Bose-Hubbard model
We study the ground state of the attractive one-dimensional Bose-Hubbard
model, and in particular the nature of the crossover between the weak
interaction and strong interaction regimes for finite system sizes. Indicator
properties like the gap between the ground and first excited energy levels, and
the incremental ground-state wavefunction overlaps are used to locate different
regimes. Using mean-field theory we predict that there are two distinct
crossovers connected to spontaneous symmetry breaking of the ground state. The
first crossover arises in an analysis valid for large L with finite N, where L
is the number of lattice sites and N is the total particle number. An
alternative approach valid for large N with finite L yields a second crossover.
For small system sizes we numerically investigate the model and observe that
there are signatures of both crossovers. We compare with exact results from
Bethe ansatz methods in several limiting cases to explore the validity for
these numerical and mean-field schemes. The results indicate that for finite
attractive systems there are generically three ground-state phases of the
model.Comment: 17 pages, 12 figures, Phys.Rev.B(accepted), minor changes and updated
reference
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