We continue the exploration of nonstandard continuum field theories related to fractons in 3+1 dimensions. Our theories exhibit exotic global and gauge symmetries, defects with restricted mobility, and interesting dualities. Depending on the model, the defects are the probe limits of either fractonic particles, strings, or strips. One of our models is the continuum limit of the plaquette Ising lattice model, which features an important role in the construction of the X-cube model.
We discuss nonstandard continuum quantum field theories in 2+1 dimensions. They exhibit exotic global symmetries, a subtle spectrum of charged excitations, and dualities similar to dualities of systems in 1+1 dimensions. These continuum models represent the low-energy limits of certain known lattice systems. One key aspect of these continuum field theories is the important role played by discontinuous field configurations. In two companion papers, we will present 3+1-dimensional versions of these systems. In particular, we will discuss continuum quantum field theories of some models of fractons.
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.
We study field theories with global dipole symmetries and gauge dipole symmetries. The famous Lifshitz theory is an example of a theory with a global dipole symmetry. We study in detail its
1+1D version with a compact field. When this global symmetry is promoted to a
U(1) dipole gauge symmetry, the corresponding gauge field is a tensor gauge field. This theory is known to lead to fractons. To resolve various subtleties in the precise meaning of these global or gauge symmetries, we place these
1+1D theories on a lattice and then take the continuum limit. Interestingly, the continuum limit is not unique. Different limits lead to different continuum theories, whose operators, defects, global symmetries, etc., are different. We also consider a lattice gauge theory with a
ZN dipole gauge group. Surprisingly, several physical observables, such as the ground state degeneracy and the mobility of defects, depend sensitively on the number of sites in the lattice. Our analysis forces us to think carefully about global symmetries that do not act on the standard Hilbert space of the theory, but only on the Hilbert space in the presence of defects. We refer to them as timelike global symmetries and discuss them in detail. These timelike global symmetries allow us to phrase the mobility restrictions of defects (including those of fractons) as a consequence of a global symmetry.
We consider XY-spin degrees of freedom on an fcc lattice, such that the system respects some subsystem global symmetry. We then gauge this global symmetry and study the corresponding
U(1) gauge theory on the fcc lattice. Surprisingly, this
U(1) gauge theory is dual to the original spin system. We also analyze a similar
ZN gauge theory on that lattice. All these systems are fractonic. The
U(1) theories are gapless and the
ZN theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics. Our
Z2 fcc gauge theory is the continuum limit of the well-known checkerboard model of fractons. Our continuum analysis leads to a straightforward proof of the known fact that this theory is dual to two copies of the
Z2 X-cube model. We find new models and new relations between known models. The
ZN fcc gauge theory can be realized by coupling three copies of an anisotropic model of lineons and planons to a certain exotic
Z2 gauge theory. Also, although for
N=2 this model is dual to two copies of the
Z2 X-cube model, a similar statement is not true for higher
N.
A bstract In recent years, the concept of global symmetry has generalized considerably. Two dramatic examples of this generalization are the exotic symmetries that govern theories with fractons and non-invertible symmetries, which do not fuse according to a group law. Only recently has the interplay between these two been examined. In this paper, we provide further examples of the interplay in the XY plaquette model, XY cube model, 1+1 d theory with global dipole symmetry, and the 2+1 d Lifshitz theory. They are analogs of the duality symmetries in 2d CTFs and are constructed by first gauging a finite subgroup of the momentum symmetry on half of spacetime and then performing a duality transformation. We analyze the fusion rules of the symmetries and find that they are condensation defects from an analog of higher gauging exotic symmetries. We also address their dependence on the UV cutoff when relevant.