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arXiv:2004.00015 · SciPost Physics

Exotic $U(1)$ symmetries, duality, and fractons in 3+1-dimensional quantum field theory

Nathan Seiberg B, Shu-Heng Shao

Published Oct 5, 2020• 184 citations
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Abstract

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.

DOI
10.21468/scipostphys.9.4.046

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Exotic U(1)U(1)U(1) symmetries, duality, and fractons in 3+1-dimensional quantum field theory
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 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.
Exotic ZN\mathbb{Z}_NZN​ symmetries, duality, and fractons in 3+1-dimensional quantum field theory
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 BFBFBF theories in 2+1 dimensions, which exhibit the mixed 't Hooft anomaly of ZN×ZN\mathbb{Z}_N \times \mathbb{Z}_NZN​×ZN​ subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for ZN×ZN\mathbb{Z}_N \times \mathbb{Z}_NZN​×ZN​ subsystem symmetry in one dimension higher. By extending the foliated-exotic duality in the fractonic BFBFBF 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 ϕϕϕ-theory in 2+1 dimensions -- a fractonic gapless scalar field theory, which has the 't Hooft anomaly of U(1)×U(1)U(1) \times U(1)U(1)×U(1) subsystem symmetry. The anomaly is captured by an SSPT phase for U(1)×U(1)U(1) \times U(1)U(1)×U(1) subsystem symmetry in 3+1 dimensions via the anomaly inflow mechanism. Extending the foliated-exotic duality to the ϕϕϕ-theory, we establish field correspondences in the ϕϕϕ-theory and construct the foliated ϕϕϕ-theory that is equivalent to the exotic ϕϕϕ-theory. This provides the first example of the foliated-exotic duality in gapless theories.
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.
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)U(1)U(1) gauge theory on the fcc lattice. Surprisingly, this U(1)U(1)U(1) gauge theory is dual to the original spin system. We also analyze a similar ZN{\mathbb{Z}}_{N}ZN​ gauge theory on that lattice. All these systems are fractonic. The U(1)U(1)U(1) theories are gapless and the ZN{\mathbb{Z}}_{N}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{\mathbb{Z}}_{2}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{\mathbb{Z}}_{2}Z2​ X-cube model. We find new models and new relations between known models. The ZN{\mathbb{Z}}_{N}ZN​ fcc gauge theory can be realized by coupling three copies of an anisotropic model of lineons and planons to a certain exotic Z2{\mathbb{Z}}_{2}Z2​ gauge theory. Also, although for N=2N=2N=2 this model is dual to two copies of the Z2{\mathbb{Z}}_{2}Z2​ X-cube model, a similar statement is not true for higher NNN.