Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers

Abstract

In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of "bulk" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition) of n+1D local topological orders by unitary multi-fusion n-categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal n-functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of "bulk = center". In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical structure which can be summarized by the functoriality of the bulk (or center). This functoriality also provides the physical meanings of some well-known mathematical results on fusion 1-categories. This work can also be viewed as the first step towards a systematic study of the category of local topological orders, and the boundary-bulk relation actually provides a useful tool for this study.

One dimensional gapped quantum phases and enriched fusion categories

“…We also show that the observables on the boundaries of these phases form an enriched category such that the boundary-bulk relation holds, i.e. the bulk is the center of a boundary [KWZ15,KWZ17]. This study proves an earlier proposal in [KZ21,KZ20b] that the enriched-categorical description works for all topological orders, SPT/SET orders and symmetry-breaking gapped phases and certain gapless phases.…”

“…This dissatisfaction motivated a new description of SPT/SET orders without gauging the symmetry [KLWZZ20a]. This description is based on the idea of boundary-bulk relation [KWZ15,KWZ17]. More precisely, an anomaly-free nd SPT/SET order should have a trivial n+1d bulk, i.e.…”

Categorical computation

“…It depicts a (potentially unstable) anomaly-free 1d (spatial dimension) topological order, together with a 0d boundary. By the mathematical theory of topological order (see for example [KWZ15]), the 0d boundary can be mathematically described by a pair (X, x), where X is a finite semisimple 1-category 2 and x is an object in X. Physically, this x is a particle-like defect located at the boundary.…”

“…By the boundary-bulk relation [KWZ15,KWZ17], the 1d topological order in the first picture in Figure 1 can be described by the 1-category of 1-functors from X to X, denoted by Fun(X, X). Objects F, G in Fun(X, X) are particle-like topological defects in this 1d topological order.…”

“…C. Physically, it means that the multi-fusion 1-category Fun(X, X) describes an unstable phase, which can flow to a stable one if we introduce certain perturbation to the phase [KWZ15]. In this case, this unstableness demands fine tuning, which makes the physical realization not fault tolerant.…”