Field Theories for type-II fractons

TL;DR

This paper develops an effective field theory for type-II fractons derived from the Haah code, highlighting the necessity of constraints for physical states and exploring a Chern-Simons-like continuum theory without constraints.

Contribution

It introduces a novel continuum field theory for type-II fractons that incorporates constraints from the lattice model and proposes a Chern-Simons-like theory without such constraints.

Findings

Effective topological theory requires a state-selection constraint.

Without the constraint, the theory describes only type-I fractons.

A Chern-Simons-like continuum theory without constraints is proposed.

Abstract

We derive an effective field theory for a type-II fracton starting from the Haah code on the lattice. The effective topological theory is not given exclusively in terms of an action; it must be supplemented with a condition that selects physical states. Without the constraint, the action only describes a type-I fracton. The constraint emerges from a condition that cube operators multiply to the identity, and it cannot be consistently implemented in the continuum theory at the operator level, but only in a weaker form, in terms of matrix elements of physical states. Informed by these studies and starting from the opposite end, i.e., the continuum, we discuss a Chern-Simons-like theory that does not need a constraint or projector, and yet has no mobile excitations. Whether this continuum theory admits a lattice counterpart remains unanswered.