The order structure of topological -algebras of unbounded operators I

In this paper we begin to study the order structure of topological -algebras of unbounded operators in Hilbert space with the investigation of the normality and the bounded decomposition property of the cones. We prove that for a large class of topological -algebras the normality of the wedge of positive elements is necessary and sufficient for a topological -algebra to be algebraically and topologically isomorphic to a -algebra of unbounded operators equipped with the uniform topology. From this theorem we obtain some corollaries, so for instance, well-known results of Lassner, Brooks and Grothendieck.     

Subalgebras of infinite C*-algebras with finite Watatani indices I. Cuntz algebras

Using fusion rules of sectors as a working hypothesis, we construct endomorphisms of the Cuntz algebra whose images have finite Watatani indices. Quasi-free KMS states on appear in a natural way associated with the endomorphisms, and we determine the Murray-von Neumann-Connes types of their GNS representations.Dedicated to Huzihiro Araki     

Dynamics of topological multipoles: I. High-order nonparaxial singular beams

Approximate boundary conditions for matching the high-order nonparaxial and paraxial singular beams propagating through free space or through a homogeneous isotropic medium are formulated. It is found that the nonparaxial and paraxial beams have almost the same structure both in the vicinity of the focal caustic and away from it. In the intermediate zone, the beam profiles are distorted, and a phase mismatch arises and increases with increasing mode order. The combined nonparaxial singular beams are devised, and their dynamics is studied. It is shown that the problem of matching the paraxial and nonparaxial wave functions may be reformulated as a problem of the phase matching for nonparaxial wave function. Such a problem is similar to the mode-dispersion problem and is unsolvable in terms of a linear problem for the entire optical axis. Thus, it makes sense to discuss either the precise matching of the paraxial and nonparaxial beams within small spatial zones or an approximate matching away from the focal plane but at long distances along the optical axis.     

Three-dimensional topological field theory and symplectic algebraic geometry I

We study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space X (the Rozansky–Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in X   equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the Z₂${\mathbb{Z}}_2$-graded derived category of X; it is also related to categories of matrix factorizations and a categorification of deformation quantization (quantization of symmetric monoidal categories). In Appendix B we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree.     

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. I.

A (non-commutative) generalization of the classical moment problem is formulated on arbitrary *-algebras with units. This is used to produce aC*-algebra associated with the space of test functions for quantum fields. ThisC*-algebra plays a role in theories of bounded localized observables in Hilbert space which is similar to that of the space of test functions in quantum field theories (namely it is represented in Hilbert space). The case of local quantum fields which satisfy a slight generalization of the growth condition is investigated. Laboratorie associé au Centre National de la Recherche Scientifique.     

(3+1)-TQFTs and topological insulators

Levin-Wen models are microscopic spin models for topological phases of matter in (2+1)-dimension. We introduce a generalization of such models to (3 + 1)-dimension based on unitary braided fusion categories, also known as unitary premodular categories. We discuss the ground state degeneracy on 3-manifolds and statistics of excitations which include both points and defect loops. Potential connections with recently proposed fractional topological insulators and projective ribbon permutation statistics are described.     

Scattering theory for quasi-free time automorphisms of C1-algebras and von Neumann algebras

Relations between scattering theory in L² space, for quasi-free automorphism groups of C¹-algebras and for those of their representations are established and it is shown that wave operators as time limits do not exist in these representations. Finally, the connection of scattering theory and dynamical stability is discussed.     

Interface-induced states at the boundary between a 3D topological insulator and a normal insulator

We show that, when a three-dimensional (3D) narrow-gap semiconductor with inverted band gap (“topological insulator,” TI) is attached to a 3D wide-gap semiconductor with non-inverted band gap (“normal insulator,” NI), two types of bound electron states having different spatial distributions and spin textures arise at the TI/NI interface. Namely, the gapless (“topological”) bound state can be accompanied by the emergence of the gapped (“ordinary”) bound state. We describe these states in the framework of the envelope function method using a variational approach for the energy functional; their existence hinges on the ambivalent character of the constraint for the envelope functions that correspond to the “open” or “natural” boundary conditions at the interface. The properties of the ordinary state strongly depend on the effective interface potential, while the topological state is insensitive to the interface potential variation.     

Graded constraint algebras,OSP(1, 1|2) and superstring theory

Earlier, we have established that, for a constrained system with a first class bosonic constraint algebra, the standard BRST invariance generalizes to an OSP(1, 1|2) symmetry, with four nilpotent and anticommuting BRST-type operators. Here we generalize this to arbitrary constrained systems with a graded first class constraint algebra. Our approach is based on the Fradkin- Vilkovisky formalism and uses a relation between abelian and nonabelian constraint algebras. Subsidiary constraints and generalized structure constants play an important role in the construction. As an application, we construct the OSP(1, 1|2) generators for superstrings. Here the subsidiary constraints are identified with physically relevant operators used in the unitarity proof.     

VI. Dynamical and topological considerations of quark confinement

Non-Abelian magnetic superconductivity provides a realisation of the topological algebra of 't Hooft which does confine quarks. Its dynamical and topological aspects are discussed and the related phenomenological theory of quark confinement is sketched.     

Symmetry protected topological phases in spin-1 ladders and their phase transitions

We study two-legged spin-1 ladder systems with D₂×σ$D_2\times \sigma$ symmetry group, where D₂$D_2$ is discrete spin rotational symmetry and σ$\sigma$ means interchain reflection symmetry. The system has one trivial phase and seven nontrivial symmetry protected topological (SPT) phases. We construct Hamiltonians to realize all of these SPT phases and study the phase transitions between them. Our numerical results indicate that there is no direct continuous transition between any two SPT phases we studied. We interpret our results via topological nonlinear sigma model effective field theory, and further conjecture that generally there is no direct continuous transition between two SPT phases in one dimension if the symmetry group is discrete at all length scales.     

Quantum criticality between topological and band insulators in 3+1 dimensions

Four-component massive and massless Dirac fermions in the presence of long range Coulomb interaction and chemical potential disorder exhibit striking fermionic quantum criticality. For an odd number of flavors of Dirac fermions, the sign of the Dirac mass distinguishes the topological and the trivial band insulator phases, and the gapless semimetallic phase corresponds to the quantum critical point that separates the two. Up to a critical strength of disorder, the semimetallic phase remains stable, and the universality class of the direct phase transition between two insulating phases is unchanged. Beyond the critical strength of disorder the semimetallic phase undergoes a phase transition into a disorder controlled diffusive metallic phase, and there is no longer a direct phase transition between the two types of insulating phases.     

Preparation and characterisation of compositionally graded BaxSr1-xTiO3 thin films

Ba_xSr_1-xTiO₃ thin films with a compositional gradient of x=0.3 to 1 (in 0.1 mole fraction increments) were fabricated on Pt/Ti/SiO₂/Si substrates using a modified sol–gel technique. The graded film crystallised in a perovskite structure and consists of a uniform microstructure with comparatively larger grains. The room-temperature relative dielectric constant (ε_r) and dielectric loss (cosδ) at 100 kHz were found to be 305 and 0.03 respectively. Dielectric peaks were not observed in the temperature range from -20 °C to 120 °C. The dielectric constant and dielectric loss were almost independent of temperature. Polarisation–electric field measurements at room temperature revealed a saturated but slim hysteresis loop with a remanent polarisation (P_r) of 0.6 μC/cm² and a coercive field (E_c) of 2.4 kV/mm. The graded film behaves as a stack of Ba_xSr_1-xTiO₃ capacitors connected in series and hence the dielectric Curie peaks are removed. Received: 4 October 2001 / Accepted: 17 October 2001 / Published online: 23 January 2002     

The 1D Ising model and the topological phase of the Kitaev chain

It has been noted that the Kitaev chain, a p-wave superconductor with nearest-neighbor pairing amplitude equal to the hopping term Δ=t$\Delta =t$, and chemical potential μ=0$\mu =0$, can be mapped into a nearest neighbor Ising model via a Jordan–Wigner transformation. Starting from the explicit eigenstates of the open Kitaev chain in terms of the original fermion operators, we elaborate that despite this formal equivalence the models are physically inequivalent, and show how the topological phase in the Kitaev chain maps into conventional order in the Ising model.     

2+1-D topological quantum field theory and 2-D conformal field theory

We describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory. Some Applications to quantum knot invariants leading to the equivalence of Chern-Simons-Witten and Kohno's approaches are outlined.     

Braiding matrices,modular transformations and topological field theories in 2+1 dimensions

Relations between 3D topological field theories and rational conformal field theories are discussed. In the former framework, we can define the generalized Verlinde operators. Using these operators, we find modular transformations for conformal blocks of one point functions and two point functions on the torus. The result is generalized to higher genus. The correctness of our formulae is illustrated by some examples. We also emphasize the importance of the fusion algebra.Addresses after October 1, 1989: Institute of Theoretical Physics, Academia Sinica, Beijing, P. R. China     

Optical properties and electronic structure of simple cubic and f.c.c. T1I

T1I-layers with simple cubic and f.c.c. structure have been prepared by evaporation of CsI and RbI. The optical absorption spectra of these layers have been measured for the first time and are shown to be very similar to those of simple cubic and f.c.c. T1C1 and T1Br, respectively. Relativistic band-structure calculations explain the optical spectra as well as these similarities.