arXiv subjects

Chris Bourne

Publications and source records attributed to Chris Bourne.

16 recordsLinked to original sources

arXiv2026

Analytic index theory and spectral flow in real Hilbert $C^*$-modules

We consider the analytic index and spectral flow of Fredholm operators on Hilbert $C^*$-modules. Our spaces and algebras are equipped with a real structure, so the analytic index and spectral flow takes value in the real $K$-theory group of a $\sigma$-unital $C^*$-algebra. We use Van Daele $K$-theory, which allows us to treat the eight real $K$-theory groups and the two complex groups on an equal footing. We provide a general definition of the analytic index for Clifford anti-linear and skew-adjoint Fredholm operators as well as self-adjoint and odd Fredholm operators. Our definition of spectral flow and its basic properties are valid for Wahl-continuous paths of Fredholm operators on a real Hilbert $C^*$-module. We also provide an analytic approach to the spectral flow as a decomposition into a finite sum of relative indices. Furthermore, we prove a real version of the Robbin-Salamon theorem, relating the spectral flow to a Fredholm index. Our description of the index and spectral flow relies on various isomorphisms between Kasparov's $KKR$-theory and Van Daele $K$-theory, which we systematically describe in the Appendix.

math.OA↗

arXiv2025

Interfaces of discrete systems - spectral and index properties

We develop a general mathematical framework to study mixtures of different physical systems brought together on a discrete interface. Adapting work by M\u{a}ntoiu et al., we use an operator algebraic framework such that the bulk systems at infinity of the mixture are recovered via the spatial asymptotics of the operators on the interface. Fixing an asymptotics and interface algebra, we show how the essential spectrum and topological properties can be inferred from the bulk systems at infinity. By working with Hilbert $C^*$-modules, we can further refine these results with respect to an ambient algebra of observables.

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arXiv2023

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.

math.OA↗

arXiv2022

Locally equivalent quasifree states and index theory

We consider quasifree ground states of Araki's self-dual CAR algebra from the viewpoint of index theory and symmetry protected topological (SPT) phases. We first review how Clifford module indices characterise a topological obstruction to connect pairs of symmetric gapped ground states. This construction is then generalised to give invariants in $KO_\ast(A^\mathfrak{r})$ with $A$ a $C^{*,\mathfrak{r}}$-algebra of allowed deformations. When $A=C^*(X)$, the Roe algebra of a coarse space $X$, and we restrict to gapped ground states that are locally equivalent with respect $X$, a $K$-homology class is also constructed. The coarse assembly map relates these two classes and clarifies the relevance of $K$-homology to free-fermionic SPT phases.

math-ph↗

arXiv2021

Localised module frames and Wannier bases from groupoid Morita equivalences

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.

math.OA↗

arXiv2021

The classification of symmetry protected topological phases of one-dimensional fermion systems

We introduce an index for symmetry protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group $G$. This index takes values in $\mathbb{Z}_2 \times H^1(G,\mathbb{Z}_2) \times H^2(G, U(1)_{\mathfrak{p}})$ with a generalized Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.

math-ph↗

arXiv2020

The Cayley transform in complex, real and graded $K$-theory

We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele $K$-theory to $KK$-theory for graded $C^*$-algebras with a real structure. Isomorphisms between $KK$-theory and complex or real $K$-theory for ungraded $C^*$-algebras are a special case of this map. In all cases our map is compatible with the computational techniques required in physical and geometrical applications, in particular index pairings and Kasparov products. We provide applications to real $K$-theory and topological phases of matter.

math.KT↗

arXiv2020

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.

math.KT↗

arXiv2020

On $\mathbb{Z}_2$-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.

math-ph↗

arXiv2019

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.

math.OA↗

arXiv2018

Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases

In order to study continuous models of disordered topological phases, we construct an unbounded Kasparov module and a semifinite spectral triple for the crossed product of a separable $C^*$-algebra by a twisted $\mathbb{R}^d$-action. The spectral triple allows us to employ the non-unital local index formula to obtain the higher Chern numbers in the continuous setting with complex observable algebra. In addition, the pairing can be extended to a larger algebra closely related to dynamical localisation, as in the tight-binding approximation. The Kasparov module allows us to exploit the Wiener-Hopf extension and the Kasparov product to obtain a bulk-boundary correspondence for continuous models of disordered topological phases.

math-ph↗

arXiv2018

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.

math-ph↗

arXiv2016

The $K$-theoretic bulk-edge correspondence for topological insulators

We study the application of Kasparov theory to topological insulator systems and the bulk-edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real $C^*$-algebras and $KKO$-theory must be used.

math-ph↗

arXiv2016

Application of semifinite index theory to weak topological phases

Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov's $KK$-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Real invariants and the bulk-boundary correspondence are also briefly considered.

math-ph↗

arXiv2016

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.

math-ph↗

arXiv2015

The bulk-edge correspondence for the quantum Hall effect in Kasparov theory

We prove the bulk-edge correspondence in $K$-theory for the quantum Hall effect by constructing an unbounded Kasparov module from a short exact sequence that links the bulk and boundary algebras. This approach allows us to represent bulk topological invariants explicitly as a Kasparov product of boundary invariants with the extension class linking the algebras. This paper focuses on the example of the discrete integer quantum Hall effect, though our general method potentially has much wider applications.

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