A Modified Villain Formulation of Fractons and Other Exotic Theories

Abstract

We reformulate known exotic theories (including theories of fractons) on a Euclidean spacetime lattice. We write them using the Villain approach and then we modify them to a convenient range of parameters. The new lattice models are closer to the continuum limit than the original lattice versions. In particular, they exhibit many of the recently found properties of the continuum theories including emergent global symmetries and surprising dualities. Also, these new models provide a clear and rigorous formulation to the continuum models and their singularities. In appendices, we use this approach to review well-studied lattice models and their continuum limits. These include the XY-model, the ${\mathbb{Z}}_N$ clock-model, and various gauge theories in diverse dimensions. This presentation clarifies the relation between the condensed-matter and the high-energy views of these systems. It emphasizes the role of symmetries associated with the topology of field space, duality, and various anomalies.

66 pages, 1 figure

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Cited by in corpus (29)

• Non-Invertible Duality Defects in 3+1 Dimensions
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• The low-energy limit of some exotic lattice theories and UV/IR mixing
• Anomaly Inflow for Subsystem Symmetries
• Dipole condensates in tilted Bose-Hubbard chains
• Exact emergent higher-form symmetries in bosonic lattice models
• Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions
• 2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Hydrodynamics of dipole-conserving fluids
• World-Line Description of Fractons
• Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities
• Boson-fermion duality with subsystem symmetry
• Fractonic plaquette-dimer liquid beyond renormalization
• Gapped Lineon and Fracton Models on Graphs
• Fractons, non-Riemannian Geometry, and Double Field Theory
• Phase structure of self-dual lattice gauge theories in 4d
• Four-fermion deformations of the massless Schwinger model and confinement
• Scalar, fermionic and supersymmetric field theories with subsystem symmetries in d+1 dimensions
• Polyakov's confinement mechanism for generalized Maxwell theory
• Villain model with long-range couplings
• Lattice regularizations of $\theta$ vacua: Anomalies and qubit models
• Anomalous Continuous Translations
• Interacting Fractons in 2+1-Dimensional Quantum Field Theory
• Fractonic Berezinskii-Kosterlitz-Thouless transition from a renormalization group perspective
• Instanton Density Operator in Lattice QCD from Higher Category Theory
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Higgs Phases and Boundary Criticality
• Z3 gauge theory coupled to fermions and quantum computing