Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory

TL;DR

This paper proposes a K-theoretic classification of anyonic topological order in 2D semi-metals, connecting topological phases, interactions, and braid statistics through twisted equivariant differential K-theory.

Contribution

It introduces a novel application of twisted equivariant differential K-theory to classify interacting anyonic topological phases in 2D semi-metals, extending beyond non-interacting classifications.

Findings

Classifies 2D semi-metal phases via flat differential twisted equivariant K-theory.

Describes n-electron phases using K-theory of configuration spaces.

Links twisting by local systems to effective gauge interactions and anyonic statistics.

Abstract

While the classification of non-interacting crystalline topological insulator phases by equivariant K-theory has become widely accepted, its generalization to anyonic interacting phases -- hence to phases with topologically ordered ground states supporting topological braid quantum gates -- has remained wide open. On the contrary, the success of K-theory with classifying non-interacting phases seems to have tacitly been perceived as precluding a K-theoretic classification of interacting topological order; and instead a mix of other proposals has been explored. However, only K-theory connects closely to the actual physics of valence electrons; and self-consistency demands that any other proposal must connect to K-theory. Here we provide a detailed argument for the classification of symmetry protected/enhanced su(2)-anyonic topological order, specifically in interacting 2d semi-metals, by the twisted equivariant differential (TED) K-theory of configuration spaces of points in the complement of nodal points inside the crystal's Brillouin torus orbi-orientifold. We argue, in particular, that: (1) topological 2d semi-metal phases modulo global mass terms are classified by the flat differential twisted equivariant K-theory of the complement of the nodal points; (2) n-electron interacting phases are classified by the K-theory of configuration spaces of n points in the Brillouin torus; (3) the somewhat neglected twisting of equivariant K-theory by "inner local systems" reflects the effective "fictitious" gauge interaction of Chen, Wilczeck, Witten & Halperin (1989), which turns fermions into anyonic quanta; (4) the induced su(2)-anyonic topological order is reflected in the twisted Chern classes of the interacting valence bundle over configuration space, constituting the hypergeometric integral construction of monodromy braid representations.