Fracton topological order from Higgs and partial confinement mechanisms of rank-two gauge theory

TL;DR

This paper explores how Higgs and partial confinement mechanisms in rank-2 gauge theories can generate fracton topological order, unifying different models and clarifying their physical origins.

Contribution

It identifies the specific mechanisms needed to produce fully gapped fracton phases from rank-2 ${ m U}(1)$ gauge theories, connecting them to known models like the X-cube.

Findings

Higgs and partial confinement mechanisms lead to fracton order

Rank-2 $ ext{Z}_2$ scalar charge theory is equivalent to four toric codes

Checkerboard fracton model viewed as a rank-2 $ ext{Z}_2$ gauge theory

Abstract

Fractons are gapped point-like excitations in $d=3$ topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor ${\rm U}(1)$ gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 ${\rm U}(1)$ scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 $\mathbb{Z}_2$ scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the $d=3$ toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 $\mathbb{Z}_2$ gauge theory with two different Gauss' law constraints on different lattice sites.