Disordered Topological Insulators via C∗C^*-Algebras

The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z_2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z_2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.Comment: Added detail regarding the mapping of almost commuting unitary matrices to almost commuting Hermitian matrices that form an approximate representation of the sphere. 6 pages, 6 figure

Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect

For models of non-interacting fermions moving within sites arranged on a surface in three dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are $K$-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward, and does not require translational invariance or introducing a flux torus. We further show that there is a $Z_2$ index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices, and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time reversal or particle-hole conjugation. Finally, in the case of the sphere--mathematically speaking three almost commuting Hermitians whose sum of square is near the identity--we give the first quantitative result showing this index is the only obstruction to finding commuting approximations. We review the known non-quantitative results for the torus.Comment: 35 pages, 2 figure

Almost commuting unitary matrices related to time reversal

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either the matrices over A for A the real numbers, A the complex numbers or A the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple. We consider two-dimensional geometries and so this obstruction lives in KO_{-2}(A). This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.Comment: 33 pages, 2 figures. In version 2 some formulas have been corrected and some proofs have been rewritten to improve the expositio

Effects of perturbative exchanges in a QCD-string model

The QCD-string model for baryons derived by Simonov and used for the calculation of baryon magnetic moments in a previous paper is extended to include also perturbative gluon and meson exchanges. The mass spectrum of the baryon multiplet is studied. For the meson interaction either the pseudoscalar or pseudovector coupling is used. Predictions are compared with the experimental data. Besides these exchanges the influence of excited quark orbitals on the baryon ground state are considered by performing a multichannel calculation. The nucleon-Delta splitting increases due to the mixing of higher quark states while the baryon magnetic momenta decrease. The multichannel calculation with perturbative exchanges is shown to yield reasonable magnetic moments while the mass spectrum is close to experiment.Comment: 37 pages Revtex with 2 figures, to be published in Phys. Atom. Nucl. dedicated to the 70th Birthday of Yu. A. Simono

Children’s learning from a Smokefree sports programme: implications for health education

Objective: This article reports on a qualitative evaluation of the Love Life, Smokefree Sports primary school pilot intervention. This 8-week programme used sports and physical activity sessions to convey Smokefree messages to 120 children aged 10 and 11 in two primary schools in Sheffield in 2018. The study aimed to understand children’s experiences of participating in the programme. Its objectives were to explore children’s recall of the health promotion messages associated with each of the learning sessions; explore children’s perceptions of the meaningfulness of those messages in the context of their everyday lives; and identify and understand any contextual factors that might impact upon children’s recall and/or the meaningfulness of the Smokefree messages. Method: Qualitative data were generated with 25 children through focus groups after the programme concluded. Data were analysed thematically using cross-sectional, categorical indexing. Results: Learning from the programme was particularly likely to be described as meaningful by children when they could interact with material and visual representations of complex ideas and when sessions involved strongly embodied experiences. However, children did not always find it easy to relate learning to their everyday lives and sometimes struggled to reconcile pre-existing, contextualised understandings with intervention messages. We mobilise the concept of critical health literacy as a theoretical lens through which to interpret these findings. Conclusion: Health education should be meaningful in the context of children’s everyday lives. Starting from the premise that children are active critical health literacy practitioners and working with them to design and evaluate health education initiatives can promote this

Almost commuting self-adjoint matrices --- the real and self-dual cases

We show that a pair of almost commuting self-adjoint, symmetric matrices is close to a pair of commuting self-adjoint, symmetric matrices (in a uniform way). Moreover we prove that the same holds with self-dual in place of symmetric. The notion of self-dual Hermitian matrices is important in physics when studying fermionic systems that have time reversal symmetry. Since a symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin's famous theorem on almost commuting matrices. Similarly the self-dual case gives a version for matrices over the quaternions. We prove analogous results for element of real C^*-algebras of "low rank." In particular, these stronger results apply to paths of almost commuting Hermitian matrices that are real or self-dual. Along the way we develop a theory of semiprojectivity for real C^*-algebras.Comment: Expanded references. 33 page