Fracton topological order via coupled layers

TL;DR

This paper introduces a coupled layer construction method for fracton topological orders in three dimensions, enabling the creation and analysis of new models with restricted excitations and complex ground-state degeneracies.

Contribution

It presents a novel coupled layer approach to construct and analyze fracton phases, including new models and connections to topological quantum field theory.

Findings

Constructed the X-cube model as a strong-coupling limit of stacked toric codes.

Developed two new fracton models: a semionic X-cube and the Four Color Cube.

Demonstrated mechanisms of p-string and p-membrane condensation for phase formation.

Abstract

In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the "Four Color Cube model." The couplings considered lead to fracton topological orders via mechanisms we dub "p-string condensation" and "p-membrane condensation," in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.