Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2201.10589 · Physical review. B./Physical review. B

Global dipole symmetry, compact Lifshitz theory, tensor gauge theory, and fractons

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

Published Jul 11, 2022• 100 citations
arXivHTMLPDF

Abstract

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 $1+1\mathrm{D}$ 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 $1+1\mathrm{D}$ 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 ${\mathbb{Z}}_{N}$ 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.

DOI
10.1103/physrevb.106.045112

Related papers

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

Pranay Gorantla, Ho Tat Lam, Nathan Seiberg B +1 more

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.

arXiv:2209.10030•Physical review. B./Physical review. B•Aug 3, 2023•37 citations

More exotic field theories in 3+1 dimensions

Pranay Gorantla, Ho Tat Lam, Nathan Seiberg B +1 more

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.

arXiv:2007.04904•SciPost Physics•Nov 16, 2020•79 citations

Systematic Constructions of Fracton Theories

Djordje Radicevic

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 $\mathfrak{F}_p$ gauge theories, in which gauge transformations of rank-$k$ fields are specified by rank-$(k - p)$ gauge parameters. The $\mathbb{Z}_2$ rank-two theory of type $\mathfrak{F}_2$, 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 $\mathrm{U}(1)$ rank-two theory of type $\mathfrak{F}_2$ 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 $\mathfrak{F}_2$ theory also show gapless fracton behavior; on a cubic lattice they have a conserved $\mathrm{U}(1)$ charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to $\mathfrak{F}_2$ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.

arXiv:1910.06336•Oct 14, 2019•29 citations

fcc lattice, checkerboards, fractons, and quantum field theory

Pranay Gorantla, Ho Tat Lam, Nathan Seiberg +1 more

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 $N$.

arXiv:2010.16414•Physical review. B./Physical review. B•May 7, 2021•38 citations

Foliated field theories and multipole symmetries

Hiromi Ebisu, Masazumi Honda, Taiichi Nakanishi

Due to the recent studies of the fracton topological phases, the host of which deconfined quasiparticle excitations with mobility restrictions, the concept of symmetries has been updated. Focusing on one of these new symmetries, multipole symmetries, including global, dipole, and quadruple symmetries, and gauge fields associated with them, we construct new sets of ${\mathbb{Z}}_{N}\phantom{\rule{4pt}{0ex}}(2+1)$-dimensional foliated background-field (BF) theories, where BF theories of conventional topological phases are stacked in layers with couplings between them. By investigating gauge-invariant nonlocal operators, we show that our foliated BF theories exhibit unusual ground-state degeneracy depending on the system size; it depends on the greatest common divisor between $N$ and the system size. Our result provides an important connection between UV lattice models of the fracton topological phases and other unconventional ones in view of foliated field theories.

arXiv:2310.06701•Physical review. B./Physical review. B•Apr 2, 2024•28 citations

Lattice 2-group symmetries: operators, defects, and gauging

Lucas Z. Brito, Salvatore D. Pace

We construct and study lattice realizations of finite 2-group symmetries in ${2+1}$d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group $G$ and 1-form symmetry group $A$: split 2-groups with trivial Postnikov class ${[β]\in\mathcal{H}^3(G,A)}$, and central 2-groups with trivial action ${ρ\colon G\to\text{Aut}(A)}$. In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice $A$ 1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of $ρ$ and $[β]$ for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.

cond-mat.str-elCross-listedarXiv:2609.10682•Sep 11, 2026•0 citations
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.
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.
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.
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.
Due to the recent studies of the fracton topological phases, the host of which deconfined quasiparticle excitations with mobility restrictions, the concept of symmetries has been updated. Focusing on one of these new symmetries, multipole symmetries, including global, dipole, and quadruple symmetries, and gauge fields associated with them, we construct new sets of ZN(2+1){\mathbb{Z}}_{N}\phantom{\rule{4pt}{0ex}}(2+1)ZN​(2+1)-dimensional foliated background-field (BF) theories, where BF theories of conventional topological phases are stacked in layers with couplings between them. By investigating gauge-invariant nonlocal operators, we show that our foliated BF theories exhibit unusual ground-state degeneracy depending on the system size; it depends on the greatest common divisor between NNN and the system size. Our result provides an important connection between UV lattice models of the fracton topological phases and other unconventional ones in view of foliated field theories.
We construct and study lattice realizations of finite 2-group symmetries in 2+1{2+1}2+1d quantum lattice systems with finite-dimensional tensor-product Hilbert spaces. We focus on two broad classes of 2-groups with 0-form symmetry group GGG and 1-form symmetry group AAA: split 2-groups with trivial Postnikov class [β]∈H3(G,A){[β]\in\mathcal{H}^3(G,A)}[β]∈H3(G,A), and central 2-groups with trivial action ρ ⁣:G→Aut(A){ρ\colon G\to\text{Aut}(A)}ρ:G→Aut(A). In both cases, we construct symmetry operators on the full tensor-product Hilbert space that become 2-group symmetry operators when restricted to the topological subspace of the lattice AAA 1-form symmetry. While the lattice split 2-group symmetry operators are onsite, the lattice central 2-group symmetry operators are not, and can only be made onsite after introducing ancillae. We extensively explore various manifestations of ρρρ and [β][β][β] for these lattice 2-group symmetry operators and demonstrate their agreement with expectations from quantum field theory. These manifestations arise in the transformation of operators carrying symmetry charge, the structure of lattice 2-group symmetry defects, and the dual fusion 2-category symmetries obtained by gauging the lattice 2-group symmetries. We further propose families of local symmetric Hamiltonians for both classes of lattice 2-group symmetries and identify exactly solvable limits lying in phases with spontaneous 2-group symmetry breaking and nontrivial symmetry-enriched topological order. In one such limit, the gauged Hamiltonians are exactly solvable lattice realizations of the corresponding 2-group gauge theories, whose ground-state degeneracies we calculate.