Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems

The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.Comment: 44 pages, 4 figures. v2: typos corrected and references updated. v3: Minor changes, to appear in J. Phys. A (Mathematical and Theoretical

Index theory of chiral unitaries and split-step quantum walks

Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a chiral-symmetric self-adjoint operator. Split-step quantum walks provide a rich class of examples. We use the index of a pair of projections and the Cayley transform to define topological indices for chiral unitaries on both Hilbert spaces and Hilbert $C^*$-modules. In the case of the discrete time evolution of a Hamiltonian-like operator, we relate the index for chiral unitaries to the index of the Hamiltonian. We also prove a double-sided winding number formula for anisotropic split-step quantum walks on Hilbert $C^*$-modules, extending a result by Matsuzawa.Comment: v2: connections to literature updated, v3: Section 2 revised, 37 page

A noncommutative framework for topological insulators

We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of noncommutative index theory of operator algebras. In particular we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realised as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample's (possibly noncommutative) Brillouin zone.Comment: 32 pages, final versio

Index theory and topological phases of aperiodic lattices

We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.Comment: 52 pages, Section 1.6 added and other minor improvements. To appear in Annales Henri Poincar\'{e

The KO-valued spectral flow for skew-adjoint Fredholm operators

In this article we give a comprehensive treatment of a `Clifford module flow' along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO${}_{*}(\mathbb{R})$ via the Clifford index of Atiyah-Bott-Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that $$\text{spectral flow} = \text{Fredholm index}.$$ That is, we show how the KO--valued spectral flow relates to a KO-valued index by proving a Robbin-Salamon type result. The Kasparov product is also used to establish a spectral flow $=$ Fredholm index result at the level of bivariant K-theory. We explain how our results incorporate previous applications of $\mathbb{Z}/ 2\mathbb{Z}$-valued spectral flow in the study of topological phases of matter.Comment: v2: 47 pages, applications to physics expande

On ℤ2-indices for ground states of fermionic chains

For parity-conserving fermionic chains, we review how to associate $\mathbb{Z}_2$-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the $\mathbb{Z}_2$-valued spectral flow provides a topological obstruction for two systems to have the same $\mathbb{Z}_2$-index. A rudimentary definition of a $\mathbb{Z}_2$-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.Comment: v2: 61 pages, many revisions. To appear in Reviews in Mathematical Physic

Predictors of crystal methamphetamine use in a community-based sample of UK men who have sex with men.

Background Crystal methamphetamine (‘crystal meth’) use by men who have sex with men is an ongoing public health issue in the UK. We conducted a descriptive epidemiological study to characterise demographic and socio-sexual risk factors for crystal meth use in a national sample of UK MSM recruited in late 2014. Methods We used data from the 2014 Gay Men's Sex Survey (n = 16,565), an online community-based survey in the UK. We used logistic regression to relate risk factors to last-year use of crystal meth. Results In univariate models, crystal meth use was significantly associated with being between the ages of 30 and 49 (30–39, OR 2.24; 40–49, OR 2.21), living in London, having received a positive HIV test result (OR 7.37, 95% CI [6.28, 8.65]), and with higher education qualifications (1.40, [1.13, 1.75]), as well as with having multiple steady (2.15, [1.73, 2.68]) and non-steady (13.83, [10.30, 18.58]) partners with condomless anal intercourse. Relationships were similar in multivariate models, but education was no longer associated with last-year crystal meth use and lack of full-time employment was. Conclusions This analysis confirms and updates previous findings from the UK. Crystal meth use may now be more concentrated in London since previous surveys. This analysis presents novel findings regarding the association between number and sexual risk with partners and last-year meth use