Topological Phases, Fixed-Point Models, Extended TQFT, and Fault-Tolerant Quantum Computation - Tensors in Spacetime

Abstract

This thesis studies the classification of topological phases of matter in terms of tensors associated to discrete spacetimes. We show that all algebraic descriptions of topological order can be reduced to a single property, namely combinatorial topological invariance. This invariance yields equations for the tensors forming a description of a phase. We coin the notion of a topological tensor scheme (tTS) consisting of such a set of tensor-network equations. tTS formalize the whole spectrum of topological phase classifications: They can describe intrinsic bosonic topological, SPT and SET, symmetry-breaking, or fermionic topological orders, in any dimension. Further they can describe not only the bulk, but also superselection sectors of topological boundaries, anyons, twist defects, corners, and all other kinds of defects. In order to systematically talk about defects, we develop the notion of extended manifolds and cellulations. The majority of this thesis is focused on describing phases via microscopic fixed-point models, where the invariance is implemented as recellulation. However, we also show how to formulate extended TQFT as tTS, where the invariance is implemented by gluing, which can be understood as a generalization of recellulation.

In contrast to much of the established literature, we do not merely postulate that higher category theory describes topological order if we correctly implement a long list of technicalities. Instead, we derive all the structures without using any category theory, starting from a coarse and simple ansatz for topological invariance. More fine-tuned structures that closely resemble higher categorical structures are then obtained by a systematic process that we call block-diagonalization. We further investigate to what extent our ansatzes for topologically invariant fixed-point models are universal in the sense that they can emulate any other arbitrarily complicated ansatz for topological invariance. We find that the ansatzes discussed in most chapters of this thesis, which contain all established algebraic descriptions of topological order, are universal only under the condition that there exists a topological boundary. We find a new tTS, corresponding to a new fixed-point ansatz, that is universal independent of the existence of a topological boundary. This provides a promising route to reconcile microscopic fixed-point models with chiral phases of matter, which is one of the major unsettled questions of the field. An indication for this possibility is that the chiral anomaly of these models has the same geometric structure as our new tTS.

We illustrate many of the above ansatzes by concrete examples. Most notably, we look at the family of models arising from cohomology theory, including twisted gauge theories. In particular, we give an efficient systematic procedure to calculate arbitrary defects of spacetime dimensions 0, 1, and 2 in such models.

As an application, we show how to construct dynamic error-correcting codes from fixed-point path integrals, by measuring defects or 1-form symmetries of these path integrals. As an example, we demonstrate that the toric code, the subsystem toric code, as well as the recently developed CSS and honeycomb Floquet codes are secretly the same code up to microscopic equivalence. We also showcase this application by constructing two new codes, namely a Floquet version of the 3+1-dimensional toric code, as well as a dynamic code for the double-semion phase.