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:2207.00854 · SciPost Physics

A symmetry principle for gauge theories with fractons

Yuji Hirono, Minyoung You, Stephen Angus, Gil Young Cho

Published Feb 20, 2024• 26 citations
arXivHTMLPDF

Abstract

Fractonic phases are new phases of matter that host excitations with restricted mobility. We show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of continuous higher-form symmetries whose conserved charges do not commute with spatial translations. We refer to such symmetries as nonuniform higher-form symmetries. These symmetries fall within the standard definition of higher-form symmetries in quantum field theory, and the corresponding symmetry generators are topological. Worldlines of particles are regarded as the charged objects of 1-form symmetries, and mobility restrictions can be implemented by introducing additional 1-form symmetries whose generators do not commute with spatial translations. These features are realized by effective field theories associated with spontaneously broken nonuniform 1-form symmetries. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu-Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. The gauge theories have emergent nonuniform magnetic symmetries, and some of the magnetic monopoles become fractonic. We identify the ’t Hooft anomalies of the nonuniform higher-form symmetries and the corresponding bulk symmetry-protected topological phases. By this method, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions.

DOI
10.21468/scipostphys.16.2.050

Related papers

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

Fractonic matter in symmetry-enriched U(1) gauge theory

Dominic J. Williamson, Zhen Bi, Meng Cheng

This study reveals the relation between symmetry restrictions on the mobility of quasiparticles and symmetry-enriched topological order, where the actions of translation and global symmetries on quasiparticle excitations do not commute. The approach is demonstrably fruitful as one can systematically classify $U$(1) gauge theories enriched by translation and certain global symmetry and identify new examples of fractonic matter. It is conjectured that all such theories can be constructed by gauging layered symmetry-protected topological phases.

arXiv:1809.10275•Physical review. B./Physical review. B•Sep 23, 2019•86 citations

Fractonic line excitations: An inroad from three-dimensional elasticity theory

Shriya Pai, Michael Pretko

We demonstrate the existence of a fundamentally new type of excitation, fractonic lines, which are linelike excitations with the restricted mobility properties of fractons. These excitations, described using an amalgamation of higher-form gauge theories with symmetric tensor gauge theories, see direct physical realization as the topological lattice defects of ordinary three-dimensional quantum crystals. Starting with the more familiar elasticity theory, we show how theory maps onto a rank-4 tensor gauge theory, with phonons corresponding to gapless gauge modes and disclination defects corresponding to linelike charges. We derive flux conservation laws which lock these linelike excitations in place, analogous to the higher moment charge conservation laws of fracton theories. This way of encoding mobility restrictions of lattice defects could shed light on melting transitions in three dimensions. This new type of extended object may also be a useful tool in the search for improved quantum error-correcting codes in three dimensions.

arXiv:1804.01536•Physical review. B./Physical review. B•Jun 4, 2018•71 citations

Fractons from Vector Gauge Theory

Leo Radzihovsky, Michael Hermele

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.

arXiv:1905.06951•Physical Review Letters•Feb 7, 2020•70 citations

Mechanisms of Fracton Phases

Han Ma

Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.

•Oct 9, 2018•0 citations

Fractons, symmetric gauge fields and geometry

Francisco Peña-Benítez

Gapless fracton phases are characterized by the conservation of certain charges and their higher moments. These charges generically couple to higher rank gauge fields. In this paper we study systems conserving charge and dipole moment, and construct the corresponding gauge fields propagating in arbitrary curved backgrounds. The relation between the symmetries of these class of systems and spacetime transformations is discussed. In fact, we argue that higher rank symmetric gauge theories are closer to gravitational fields than to a standard gauge theory.

arXiv:2107.13884•Physical Review Research•Feb 13, 2023•40 citations
Fractonic phases are new phases of matter that host excitations with restricted mobility. We show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of continuous higher-form symmetries whose conserved charges do not commute with spatial translations. We refer to such symmetries as nonuniform higher-form symmetries. These symmetries fall within the standard definition of higher-form symmetries in quantum field theory, and the corresponding symmetry generators are topological. Worldlines of particles are regarded as the charged objects of 1-form symmetries, and mobility restrictions can be implemented by introducing additional 1-form symmetries whose generators do not commute with spatial translations. These features are realized by effective field theories associated with spontaneously broken nonuniform 1-form symmetries. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu-Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. The gauge theories have emergent nonuniform magnetic symmetries, and some of the magnetic monopoles become fractonic. We identify the ’t Hooft anomalies of the nonuniform higher-form symmetries and the corresponding bulk symmetry-protected topological phases. By this method, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions.
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.
This study reveals the relation between symmetry restrictions on the mobility of quasiparticles and symmetry-enriched topological order, where the actions of translation and global symmetries on quasiparticle excitations do not commute. The approach is demonstrably fruitful as one can systematically classify UUU(1) gauge theories enriched by translation and certain global symmetry and identify new examples of fractonic matter. It is conjectured that all such theories can be constructed by gauging layered symmetry-protected topological phases.
We demonstrate the existence of a fundamentally new type of excitation, fractonic lines, which are linelike excitations with the restricted mobility properties of fractons. These excitations, described using an amalgamation of higher-form gauge theories with symmetric tensor gauge theories, see direct physical realization as the topological lattice defects of ordinary three-dimensional quantum crystals. Starting with the more familiar elasticity theory, we show how theory maps onto a rank-4 tensor gauge theory, with phonons corresponding to gapless gauge modes and disclination defects corresponding to linelike charges. We derive flux conservation laws which lock these linelike excitations in place, analogous to the higher moment charge conservation laws of fracton theories. This way of encoding mobility restrictions of lattice defects could shed light on melting transitions in three dimensions. This new type of extended object may also be a useful tool in the search for improved quantum error-correcting codes in three dimensions.
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.
Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.
Gapless fracton phases are characterized by the conservation of certain charges and their higher moments. These charges generically couple to higher rank gauge fields. In this paper we study systems conserving charge and dipole moment, and construct the corresponding gauge fields propagating in arbitrary curved backgrounds. The relation between the symmetries of these class of systems and spacetime transformations is discussed. In fact, we argue that higher rank symmetric gauge theories are closer to gravitational fields than to a standard gauge theory.