Anyonic topological order in twisted equivariant differential (TED) K-theory

• Published in: Reviews in mathematical physics Vol. 35; no. 3
• Main Authors: Sati, Hisham, Schreiber, Urs
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.04.2023; World Scientific Publishing Co. Pte., Ltd
• Subjects: Braiding; Branes; Classification; Configurations; Crystal defects; Electrons; Fermions; Phases; String theory; System effectiveness; Topological insulators; Toruses; anyons; K-theory; Topological phases of matter; topological order
• ISSN: 0129-055X; 1793-6659
• DOI: 10.1142/S0129055X23500010

While the classification of noninteracting 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 noninteracting 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 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 and Halperin (1989), which turns fermions into anyonic quanta; (4) the induced 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. A tight dictionary relates these arguments to those for classifying defect brane charges in string theory [H. Sati and U. Schreiber, Anyonic defect branes in TED-K-theory, arXiv:2203.11838], which we expect to be the images of momentum-space 2 -anyons under a nonperturbative version of the AdS/CMT correspondence.