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arXiv:2010.16414 · Physical review. B./Physical review. B

fcc lattice, checkerboards, fractons, and quantum field theory

Pranay Gorantla, Ho Tat Lam, Nathan Seiberg, Shu-Heng Shao

Published May 7, 2021• 38 citations
arXivHTMLPDF

Abstract

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 ${\mathbb{Z}}_{N}$ gauge theory on that lattice. All these systems are fractonic. The $U(1)$ theories are gapless and the ${\mathbb{Z}}_{N}$ theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics. Our ${\mathbb{Z}}_{2}$ 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 ${\mathbb{Z}}_{2}$ X-cube model. We find new models and new relations between known models. The ${\mathbb{Z}}_{N}$ fcc gauge theory can be realized by coupling three copies of an anisotropic model of lineons and planons to a certain exotic ${\mathbb{Z}}_{2}$ gauge theory. Also, although for $N=2$ this model is dual to two copies of the ${\mathbb{Z}}_{2}$ X-cube model, a similar statement is not true for higher $N$.

DOI
10.1103/physrevb.103.205116

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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.
Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called Fp\mathfrak{F}_pFp​ gauge theories, in which gauge transformations of rank-kkk fields are specified by rank-(k−p)(k - p)(k−p) gauge parameters. The Z2\mathbb{Z}_2Z2​ rank-two theory of type F2\mathfrak{F}_2F2​, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the U(1)\mathrm{U}(1)U(1) rank-two theory of type F2\mathfrak{F}_2F2​ is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this F2\mathfrak{F}_2F2​ theory also show gapless fracton behavior; on a cubic lattice they have a conserved U(1)\mathrm{U}(1)U(1) charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to F2\mathfrak{F}_2F2​ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.
There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.
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.
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)\text{U}(1)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.
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+1D1+1\mathrm{D}1+1D version with a compact field. When this global symmetry is promoted to a U(1)U(1)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+1D1+1\mathrm{D}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{\mathbb{Z}}_{N}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.