916 research outputs found
Solutions to the Quantum Yang-Baxter Equation with Extra Non-Additive Parameters
We present a systematic technique to construct solutions to the Yang-Baxter
equation which depend not only on a spectral parameter but in addition on
further continuous parameters. These extra parameters enter the Yang-Baxter
equation in a similar way to the spectral parameter but in a non-additive form.
We exploit the fact that quantum non-compact algebras such as
and type-I quantum superalgebras such as and are
known to admit non-trivial one-parameter families of infinite-dimensional and
finite dimensional irreps, respectively, even for generic . We develop a
technique for constructing the corresponding spectral-dependent R-matrices. As
examples we work out the the -matrices for the three quantum algebras
mentioned above in certain representations.Comment: 13 page
Lattice Models
In this paper I construct lattice models with an underlying
superalgebra symmetry. I find new solutions to the graded Yang-Baxter equation.
These {\it trigonometric} -matrices depend on {\it three} continuous
parameters, the spectral parameter, the deformation parameter and the
parameter, , of the superalgebra. It must be emphasized that the
parameter is generic and the parameter does not correspond to the
`nilpotency' parameter of \cite{gs}. The rational limits are given; they also
depend on the parameter and this dependence cannot be rescaled away. I
give the Bethe ansatz solution of the lattice models built from some of these
-matrices, while for other matrices, due to the particular nature of the
representation theory of , I conjecture the result. The parameter
appears as a continuous generalized spin. Finally I briefly discuss the problem
of finding the ground state of these models.Comment: 19 pages, plain LaTeX, no figures. Minor changes (version accepted
for publication
Universal integrability objects
We discuss the main points of the quantum group approach in the theory of
quantum integrable systems and illustrate them for the case of the quantum
group . We give a complete set of the
functional relations correcting inexactitudes of the previous considerations. A
special attention is given to the connection of the representations used to
construct the universal transfer operators and -operators.Comment: 21 pages, submitted to the Proceedings of the International Workshop
"CQIS-2012" (Dubna, January 23-27, 2012
Free Dirac evolution as a quantum random walk
Any positive-energy state of a free Dirac particle that is initially
highly-localized, evolves in time by spreading at speeds close to the speed of
light. This general phenomenon is explained by the fact that the Dirac
evolution can be approximated arbitrarily closely by a quantum random walk,
where the roles of coin and walker systems are naturally attributed to the spin
and position degrees of freedom of the particle. Initially entangled and
spatially localized spin-position states evolve with asymptotic two-horned
distributions of the position probability, familiar from earlier studies of
quantum walks. For the Dirac particle, the two horns travel apart at close to
the speed of light.Comment: 16 pages, 1 figure. Latex2e fil
Comments on Drinfeld Realization of Quantum Affine Superalgebra and its Hopf Algebra Structure
By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to
supersymmetric cases, we obtain Drinfeld current realization for quantum affine
superalgebra . We find a simple coproduct for the quantum
current generators and establish the Hopf algebra structure of this super
current algebra.Comment: Some errors and misprints corrected and a remark in section 4
removed. 12 pages, Latex fil
Multivortex Solutions of the Weierstrass Representation
The connection between the complex Sine and Sinh-Gordon equations on the
complex plane associated with a Weierstrass type system and the possibility of
construction of several classes of multivortex solutions is discussed in
detail. We perform the Painlev\'e test and analyse the possibility of deriving
the B\"acklund transformation from the singularity analysis of the complex
Sine-Gordon equation. We make use of the analysis using the known relations for
the Painlev\'{e} equations to construct explicit formulae in terms of the
Umemura polynomials which are -functions for rational solutions of the
third Painlev\'{e} equation. New classes of multivortex solutions of a
Weierstrass system are obtained through the use of this proposed procedure.
Some physical applications are mentioned in the area of the vortex Higgs
model when the complex Sine-Gordon equation is reduced to coupled Riccati
equations.Comment: 27 pages LaTeX2e, 1 encapsulated Postscript figur
Quantum Mechanics as an Approximation to Classical Mechanics in Hilbert Space
Classical mechanics is formulated in complex Hilbert space with the
introduction of a commutative product of operators, an antisymmetric bracket,
and a quasidensity operator. These are analogues of the star product, the Moyal
bracket, and the Wigner function in the phase space formulation of quantum
mechanics. Classical mechanics can now be viewed as a deformation of quantum
mechanics. The forms of semiquantum approximations to classical mechanics are
indicated.Comment: 10 pages, Latex2e file, references added, minor clarifications mad
On the solution of a supersymmetric model of correlated electrons
We consider the exact solution of a model of correlated electrons based on
the superalgebra . The corresponding Bethe ansatz equations have an
interesting form. We derive an expression for the ground state energy at half
filling. We also present the eigenvalue of the transfer matrix commuting with
the Hamiltonian.Comment: Palin latex , 8 page
Semiquantum versus semiclassical mechanics for simple nonlinear systems
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations
Transfer matrix eigenvalues of the anisotropic multiparametric U model
A multiparametric extension of the anisotropic U model is discussed which
maintains integrability. The R-matrix solving the Yang-Baxter equation is
obtained through a twisting construction applied to the underlying Uq(sl(2|1))
superalgebraic structure which introduces the additional free parameters that
arise in the model. Three forms of Bethe ansatz solution for the transfer
matrix eigenvalues are given which we show to be equivalent.Comment: 26 pages, no figures, LaTe
- …