Cyclic cocycles and quantized pairings in materials science

TL;DR

This paper explores the topological invariants derived from cyclic cohomology and K-theory in materials science, demonstrating their stability in Sobolev domains through simulations and extending the theory to many-fermion systems.

Contribution

It reviews recent developments in the index theory of cyclic cocycles for many-fermion systems, extending existing single-particle results to more complex correlated systems.

Findings

Cyclic cocycles relate to physical response coefficients in materials.

Simulations show invariants persist in Sobolev domains.

New theoretical framework for many-fermion systems derived.

Abstract

The pairings between the cyclic cohomologies and the K-theories of separable $C^\ast$-algebras supply topological invariants that often relate to physical response coefficients of materials. Using three numerical simulations, we exemplify how some of these invariants survive throughout the full Sobolev domains of the cocycles. These interesting phenomena, which can be explained by index theorems derived from Alain Connes' quantized calculus, are now well understood in the independent electron picture. Here, we review recent developments addressing the dynamics of correlated many-fermions systems, obtained in collaboration with Bram Mesland. They supply a complete characterization of an algebra of relevant derivations over the $C^\ast$-algebra of canonical anti-commutation relations indexed by a generic discrete Delone lattice. It is argued here that these results already supply the means to generate interesting and relevant states over this algebra of derivations and to identify the cyclic cocycles corresponding to the transport coefficients of the many-fermion systems. The existing index theorems for the pairings of these cocycles, in the restrictive single fermion setting, are reviewed and updated with an emphasis on pushing the analysis on Sobolev domains. An assessment of possible generalizations to the many-body setting is given.