We extend our exploration of nonstandard continuum quantum field theories in 2+1 2 + 1 dimensions to 3+1 3 + 1 dimensions. These theories exhibit exotic global symmetries, a peculiar spectrum of charged states, unusual gauge symmetries, and surprising dualities. Many of the systems we study have a known lattice construction. In particular, one of them is a known gapless fracton model. The novelty here is in their continuum field theory description. In this paper, we focus on models with a global U(1) U ( 1 ) symmetry and in a followup paper we will study models with a global \mathbb{Z}_N ℤ N symmetry.
There has been proposed two continuum descriptions of fracton systems: foliated quantum field theories (FQFTs) and exotic quantum field theories. Certain fracton systems are believed to admit descriptions by both, and hence a duality is expected between such a class of FQFTs and exotic QFTs. In this paper we study this duality in detail for concrete examples in 2+1 2 + 1 and 3+1 3 + 1 dimensions. In the examples, both sides of the continuum theories are of BF B F -type, and we find the explicit correspondences of gauge-invariant operators, gauge fields, parameters, and allowed singularities and discontinuities. This deepens the understanding of dualities in fractonic quantum field theories.
The (2+1)-dimensional continuum Lifshitz theory of a free compact scalar field plays a prominent role in a variety of quantum systems in condensed matter physics and high energy physics. It is known that in compact space, it has an infinite ground-state degeneracy. In order to understand this theory better, we consider two candidate lattice regularizations of it using the modified Villain formalism. We show that these two lattice theories have significantly different global symmetries (including a dipole global symmetry), anomalies, ground-state degeneracies, and dualities. In particular, one of them is self-dual. Given these theories and their global symmetries, we can couple them to corresponding gauge theories. These are two different
U(1) tensor gauge theories. The resulting models have excitations with restricted mobility, i.e., fractons. Finally, we give an exact lattice realization of the fracton and lineon-elasticity dualities for the Lifshitz theory, and scalar and vector charge gauge theories.
Fracton phases are new types of phases of matter characterized by subsystem global symmetry, which is a generalized global symmetry whose symmetry operator is partially topological. Their continuum low-energy effective descriptions admit two different formulations: an exotic quantum field theory (QFT) using exotic tensor gauge fields, and a foliated QFT constructed from a foliation structure and foliated gauge fields. For certain fracton QFTs, these two descriptions are equivalent, which is called the foliated-exotic duality. In this dissertation, we extend the foliated-exotic duality by combining it with the anomaly inflow mechanism for 't Hooft anomalies of subsystem symmetries. This dissertation has two main results. First, we discuss the exotic and foliated
BF theories in 2+1 dimensions, which exhibit the mixed 't Hooft anomaly of
ZN×ZN subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for
ZN×ZN subsystem symmetry in one dimension higher. By extending the foliated-exotic duality in the fractonic
BF theory to the SSPT phase, we establish the field correspondences in the SSPT phase and construct the foliated description of the SSPT phase. Second, we discuss the exotic
ϕ-theory in 2+1 dimensions -- a fractonic gapless scalar field theory, which has the 't Hooft anomaly of
U(1)×U(1) subsystem symmetry. The anomaly is captured by an SSPT phase for
U(1)×U(1) subsystem symmetry in 3+1 dimensions via the anomaly inflow mechanism. Extending the foliated-exotic duality to the
ϕ-theory, we establish field correspondences in the
ϕ-theory and construct the foliated
ϕ-theory that is equivalent to the exotic
ϕ-theory. This provides the first example of the foliated-exotic duality in gapless theories.