Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons

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+1d 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. In order to resolve various subtleties in the precise meaning of these global or gauge symmetries, we place these 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 ${\mathbb{Z}}_N$ 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 time-like global symmetries and discuss them in detail. These time-like global symmetries allow us to phrase the mobility restrictions of defects (including those of fractons) as a consequence of a global symmetry.

88+1 pages, 5 figures, 4 tables

References in corpus (39)

• Generalized Global Symmetries
• Symmetries and Strings in Field Theory and Gravity
• Local stabilizer codes in three dimensions without string logical operators
• Topological Order and Conformal Quantum Critical Points
• Fracton Phases of Matter
• Fracton hydrodynamics
• Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory
• Exotic $U(1)$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Field Theories With a Vector Global Symmetry
• Gapless Bosonic Excitation without symmetry breaking: Novel Algebraic Spin liquid with soft Gravitons
• A Modified Villain Formulation of Fractons and Other Exotic Theories
• The low-energy limit of some exotic lattice theories and UV/IR mixing
• The dipolar Bose-Hubbard model
• Exotic ${\mathbb{Z}}_N$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Fractonic Superfluids
• Resonating plaquette phases in large spin cold atom systems
• Multipolar Topological Field Theories: Bridging Higher Order Topological Insulators and Fractons
• Spontaneous breaking of multipole symmetries
• Anomaly Inflow for Subsystem Symmetries
• Fracton phases via exotic higher-form symmetry-breaking
• More Exotic Field Theories in 3+1 Dimensions
• Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry
• Emergent fractons and algebraic quantum liquid from plaquette melting transitions
• Fractons with Twisted Boundary Conditions and Their Symmetries
• A lattice bosonic model as a quantum theory of gravity
• Rank-2 Toric Code in Two Dimensions
• $(k,n)$-fractonic Maxwell theory
• FCC, Checkerboards, Fractons, and QFT
• Resonating singlet valence plaquettes
• Spontaneously Broken Subsystem Symmetries
• Higher-form Gauge Symmetries in Multipole Topological Phases
• Possible realization of the Exciton Bose Liquid phase in a hard-core boson model with ring-only exchange interactions
• Effective Field Theory of Dipolar Braiding Statistics in Two Dimensions
• Fractonic critical point proximate to a higher-order topological insulator: How does UV blend with IR?
• Reduced Conformal Symmetry
• A Duality Between U(1) Haah Code and 3D Smectic A Phase
• Screw dislocations in the X-cube fracton model
• Interacting Fractons in 2+1-Dimensional Quantum Field Theory

Cited by in corpus (70)

• The dipolar Bose-Hubbard model
• Carroll/fracton particles and their correspondence
• Subsystem Non-Invertible Symmetry Operators and Defects
• Fractonic Luttinger Liquids and Supersolids in a Constrained Bose-Hubbard Model
• Goldstone bosons and fluctuating hydrodynamics with dipole and momentum conservation
• Dipole condensates in tilted Bose-Hubbard chains
• Position-Dependent Excitations and UV/IR Mixing in the ${\mathbb{Z}}_N$ Rank-2 Toric Code and its Low-Energy Effective Field Theory
• Exact emergent higher-form symmetries in bosonic lattice models
• Hohenberg-Mermin-Wagner-type theorems and dipole symmetry
• 2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Effective Fractonic Behavior in a Two-Dimensional Exactly Solvable Spin Liquid
• Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities
• Generating Lattice Non-invertible Symmetries
• Foliated BF theories and Multipole symmetries
• Boson-fermion duality with subsystem symmetry
• Gauging fractons and linearized gravity
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• A Chern-Simons theory for dipole symmetry
• Foliated-Exotic Duality in Fractonic BF Theories
• A symmetry principle for gauge theories with fractons
• Aspects of ${\mathbb{Z}}_N$ rank-2 gauge theory in $(2+1)$ dimensions: construction schemes, holonomies, and sublattice one-form symmetries
• Effective Field Theory of Dipolar Braiding Statistics in Two Dimensions
• Gapped Lineon and Fracton Models on Graphs
• Deconfinement Dynamics of Fractons in Tilted Bose-Hubbard Chains
• Multipole and fracton topological order via gauging foliated SPT phases
• Hierarchical hydrodynamics in long-range multipole-conserving systems
• Exotic quantum liquids in Bose-Hubbard models with spatially-modulated symmetries
• Fractons on Graphs and Complexity
• Fractons, non-Riemannian Geometry, and Double Field Theory
• Detecting Subsystem Symmetry Protected Topological Order Through Strange Correlators
• Scalar, fermionic and supersymmetric field theories with subsystem symmetries in d+1 dimensions
• Noncommutative Field Theory of the Tkachenko Mode: Symmetries and Decay Rate
• Scale without Conformal Invariance in Dipolar Ferromagnets
• Anomalous hydrodynamics with triangular point group in 2 + 1 dimensions
• Boundary Modes in the Chamon Model
• Quantum vortex lattice: Lifshitz duality, topological defects and multipole symmetries
• Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi-Point Strange Correlators
• Anomaly inflow for dipole symmetry and higher form foliated field theories
• Asymptotic symmetries and soft charges of fractons
• Topological order and Fractons from Gauging Exponential Symmetries
• Filling constraints on translation invariant dipole conserving systems
• Noninvertible operators in one, two, and three dimensions via gauging spatially modulated symmetry
• Quantum Hydrodynamics of Fractonic Superfluids with Lineon Condensate: from Navier-Stokes-like Equations to Landau-like Criterion
• Helical superfluid in a frustrated honeycomb Bose-Hubbard model
• Ferromagnets, a New Anomaly, Instantons, and (noninvertible) Continuous Translations
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Unveiling UV/IR Mixing via Symmetry Defects: A View from Topological Entanglement Entropy
• Non-equilibrium dynamics of bosons with dipole symmetry: Large-$N$ Keldysh approach
• Fractonic Quantum Quench in Dipole-constrained Bosons
• Nonlinear Lifshitz Photon Theory in Condensed Matter Systems
• Entanglement entropy of higher rank topological phases
• Non-Invertible Gauss Law and Axions
• Gauging Modulated Symmetries via Multiple Gauge Symmetry Operators and Adaptive Quantum Circuits
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Fractonic superfluids. III. Hybridizing higher moments
• Topological Dipole Insulator
• Infinite-component $BF$ field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
• Symmetry-enriched topological order and quasifractonic behavior in ${\mathbb{Z}}_N$ stabilizer codes
• Exponential $U(1)$ Symmetry-Breaking Phase as a Disorder-Free Quantum Glass
• Non-invertible translation from Lieb-Schultz-Mattis anomaly
• Exotic theta terms in 2+1d fractonic field theory
• Constrained motions and slow dynamics in one-dimensional bosons with double-well dispersion
• Covariant Fracton Electrodynamics in Six Dimensions
• Emergent dipole field theory in atomic ladders
• Momentum-space modulated symmetries in the Luttinger liquid
• Preparing Code States via Seed-Entangler-Enriched Sequential Quantum Circuits: Application to Tetra-Digit Topological Error-Correcting Codes
• Hollow Lattice Tensor Gauge Theories with Bosonic Matter
• Matrix product state classification of 1D multipole symmetry protected topological phases