Abstract
Fracton phases exhibit striking behavior which appears to render them beyond the standard topological quantum field theory (TQFT) paradigm for classifying gapped quantum matter. Here, we explore fracton phases from the perspective of defect TQFTs and show that topological defect networks---networks of topological defects embedded in stratified 3+1D TQFTs---provide a unified framework for describing various types of gapped fracton phases. In this picture, the sub-dimensional excitations characteristic of fractonic matter are a consequence of mobility restrictions imposed by the defect network. We conjecture that all gapped phases, including fracton phases, admit a topological defect network description and support this claim by explicitly providing such a construction for many well-known fracton models, including the X-Cube and Haah's B code. To highlight the generality of our framework, we also provide a defect network construction of a novel fracton phase hosting non-Abelian fractons. As a byproduct of this construction, we obtain a generalized membrane-net description for fractonic ground states as well as an argument that our conjecture implies no type-II topological fracton phases exist in 2+1D gapped systems. Our work also sheds light on new techniques for constructing higher order gapped boundaries of 3+1D TQFTs.
Highlights
At first a singularly peculiar model displaying behavior vastly different from that expected of well-behaved quantum phases, Haah’s code [1] represents perhaps the best known example of fractonic matter: an entire family of renegade quantum phases which resist fitting neatly into existing paradigms for classifying quantum matter
While in principle subsumed under defect topological quantum field theory (TQFT), topological defect networks provide a novel framework as they are composed of an extensive network of topological defects enmeshed in a TQFT, and, as we show in this paper, are capable of describing gapped fracton phases
We have demonstrated that a comprehensive variety of gapped fracton phases can be described by topological defect networks
Summary
At first a singularly peculiar model displaying behavior vastly different from that expected of well-behaved quantum phases, Haah’s code [1] represents perhaps the best known example of fractonic matter: an entire family of renegade quantum phases which resist fitting neatly into existing paradigms for classifying quantum matter. This is especially true for topological orders in 2+1 dimensions (without any global symmetries), the classification of which in terms of modular tensor categories [60,61] is widely accepted to be complete Progress in this direction was aided in part by families of exactly solvable models [61,62] which encapsulate the universal features of long-range entangled phases and provide a general framework for studying fractionalized excitations. Besides the fact that gapped fracton phases appear to transcend TQFTs, any underlying mathematical framework is further obscured by their evolving typology; even in the restricted setting of translation invariant commuting projector Hamiltonians, there are a plethora of known examples which fall under the fractonic umbrella but differ in significant ways These models have been classified into type-I and type-II phases in the taxonomy of Ref. Drawing inspiration from the field of defect TQFTs [93,94,95,96,97,98], as well as from the recent classification of crystalline SPT phases [99,100,101], we show that topological defect networks are a unified framework for describing all types of gapped fracton phases
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