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arXiv:2209.10030 · Physical review. B./Physical review. B

(2+1)-dimensional compact Lifshitz theory, tensor gauge theory, and fractons

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

Published Aug 3, 2023• 37 citations
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Abstract

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 $\text{U}(1)$ 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.

DOI
10.1103/physrevb.108.075106

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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.
A bstract We construct Villain Hamiltonians for compact scalars and abelian gauge theories. The Villain integers are promoted to integral spectrum operators, whose canonical conjugates are naturally compact scalars. Further, depending on the theory, these conjugate operators can be interpreted as (higher-form) gauge fields. If a gauge symmetry is imposed on these dual gauge fields, a natural constraint on the Villain operator leads to the absence of defects (e.g. vortices, monopoles,…). These lattice models therefore have the same symmetry and anomaly structure as their corresponding continuum models. Moreover they can be formulated in a way that makes the well-know dualities look manifest, e.g. a compact scalar in 2d has a T-duality, in 3d is dual to a U(1) gauge theory, etc. We further discuss the gauged version of compact scalars on the lattice, its anomalies and solution, as well as a particular limit of the gauged XY model at strong coupling which reduces to the transverse-field Ising model. The construction for higher-form gauge theories is similar. We apply these ideas to the constructions of some models which are of interest to fracton physics, in particular the XY-plaquette model and the tensor gauge field model. The XY-plaquette model in 2+1d coupled to a tensor gauge fields at strong gauge coupling is also exactly described by a transverse field quantum J 1 − J 2 Ising model with J 1 = 2 J 2 , and discuss the phase structure of such models.
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.
Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fracton phases via a condensation of stringlike excitations built out of charged particles, analogous to the p-string condensation mechanism of the gapped X-cube fracton phase.
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.
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.