The symmetry-breaking pattern of scalars in low dimension representations

We discuss the minimum problem of an SU(n) symmetric potential which has number of Higgs in fundamental, adjoint, second-rank symmetric and antisymmetric tensor representations. When there is only one scalar in the adjoint or tensor representation, the surviving symmetry cannot be chosen arbitrarily small by adjusting the parameters in the potential. When there is one more adjoint or tensor representation, the surviving symmetry after symmetry breaking can be diminished to nothing. However, when the two higher dimension representation scalars are diagonalized or quasi-diagonalized simultaneously at the local minimum, the surviving symmetry exists. We give the conditions for the existence of such a local minimum and its symmetry-breaking pattern. Our method is to transfer these problems to the relative orientation of vectors and the number of different eigenvalues.     

Thermodynamic phase diagram of the quantum hall skyrmion system

We numerically study the interacting quantum Hall Skyrmion system based on the Chern-Simons action. By noticing that the action is invariant under global spin rotations in the spin space with respect to the magnetic field direction, we obtain the low-energy effective action for a many-Skyrmion system. Performing extensive molecular dynamics simulations, we establish the thermodynamic phase diagram for a many-Skyrmion system.     

Short-range interacting skyrmion charges in the long-range interacting skyrmion lattice

A topological excitation-skyrmion in p-wave superconductors is studied in the context of Landau-Ginzburg-Wilson (LGW) theory. Interaction between skyrmion charge densities is shown to be short ranged from a derived effective field model. The computed energy per single skyrmion of skyrmion lattice suggests a long ranged lattice interaction.     

The trace on projective representations of quantum groups

For certain roots of unity, we consider the categories of weight modules over three quantum groups: small, unrestricted and unrolled. The first main theorem of this paper is to show that there is a modified trace on the projective modules of the first two categories. The second main theorem is to show that category over the unrolled quantum group is ribbon. Partial results related to these theorems were known previously.     

Canonical transformations to action and angle variables and their representations in quantum mechanics: III. The general problem

The representation in quantum mechanics of canonical transformations to action and angle variables is discussed for a general class of Hamiltonians including some that have both bounded and unbounded orbits where, in the latter case, the definition of the transformation is suitably extended. Again, as in the particular examples discussed in the previous papers of this series, the canonical transformations are nonlinear and nonbijective. We can recover though the bijectiveness, i.e., the one-to-one onto mapping, either by introducing a sheet structure in the original phase space or using the concepts of ambiguity group and ambiguity spin. With the help of these concepts we obtain an expression for the representation in quantum mechanics of the canonical transformation and recover the latter when we pass to the classical limit with the help of the WKB approximation. Furthermore, we establish in this paper a one-to-one correspondance between the arbitrariness in the phase of the representation and in the choice of the variable conjugate to the action.     

Low-concentration series in general dimension

We discuss recent work on the development and analysis of low-concentration series. For many models, the recent breakthrough in the extremely efficient no- free-end method of series generation facilitates the derivation of 15th-order series for multiple moments in general dimension. The 15th-order series have been obtained for lattice animals, percolation, and the Edwards-Anderson Ising spin glass. In the latter cases multiple moments have been found. From complete graph tables through to 13th order, general dimension 13th-order series have been derived for the resistive susceptibility, the moments of the logarithms of the distribution of currents in resistor networks, and the average transmission coefficient in the quantum percolation problem, 11th-order series have been found for several other systems, including the crossover from animals to percolation, the full resistance distribution, nonlinear resistive susceptibility and current distribution in dilute resistor networks, diffusion on percolation clusters, the dilute Ising model, dilute antiferromagnet in a field, and random field Ising model and self-avoiding walks on percolation clusters. Series for the dilute spin-1/2 quantum Heisenberg ferromagnet are in the process of development. Analysis of these series gives estimates for critical thresholds, amplitude ratios, and critical exponents for all dimensions. Where comparisons are possible, our series results are in good agreement with both-expansion results near the upper critical dimension and with exact results (when available) in low dimensions, and are competitive with other numerical approaches in intermediate realistic dimensions.     

The unitary irreducible representations of the quantum heisenberg algebra

The goal of this work is to describe the irreducible representations of the quantum Heisenberg algebra and the unitary irreducible representation of one of its real forms. The solution of this problem is obtained through the investigation of theleft spectrum of the quantum Heisenberg algebra using the result about spectra of generic algebras of skew differential operators (cf. [R]).     

Operator representations on quantum spaces

In this article we present explicit formulae for q-differentiation on quantum spaces which could be of particular importance in physics, i.e., q-deformed Minkowski space and q-deformed Euclidean space in three or four dimensions. The calculations are based on the covariant differential calculus of these quantum spaces. Furthermore, our formulae can be regarded as a generalization of Jacksons q-derivative to three and four dimensions.Received: 26 September 2002, Revised: 18 June 2003, Published online: 2 October 2003     

Induced representations of quantum kinematical algebras

We construct the induced representations of the null-plane quantum Poincaré and quantum kappa Galilei algebras in (1+1) dimensions. The induction procedure makes use of the concept of module and is based on the existence of a pair of Hopf algebras with a nondegenerate pairing and dual bases.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

Neural network representations of quantum many-body states

Machine learning is currently the most active interdisciplinary field having numerous applications;additionally,machine-learning techniques are used to research quantum many-body problems.In this study,we first propose neural network quantum states(NNQSs)with general input observables and explore a few related properties,such as the tensor product and local unitary operation.Second,we determine the necessary and sufficient conditions for the representability of a general graph state using normalized NNQS.Finally,to quantify the approximation degree of a given pure state,we define the best approximation degree using normalized NNQSs.Furthermore,we observe that some 7V-qubit states can be represented by a normalized NNQS,such as separable pure states,Bell states and GHZ states.     

On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU _q (1,1) and E(2) in Schwinger's way.     

On classical representations of finite-dimensional quantum mechanics

In the case of a finite-dimensional Hilbert space, it is shown that quantum mechanics can be embedded into discrete classical probability theory. In particular, states can be represented as stochastic vectors and observables as random variables such that all probabilities and expectation values are given in classical terms.     

Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

Irreducible unitary representations of quantum Lorentz group

A complete classification of irreducible unitary representations of a one parameter deformationS _q L(2,C) (0<q<1) ofSL(2,C) is given. It shows that in spite of a popular belief the representation theory forS _q L(2,C) is not a smooth deformation of the one forSL(2,C).