Planon-modular fracton orders

Abstract

There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.

74 pages double-spaced, 11 figures, 3 tables. v2: references added

References in corpus (42)

• Anyons in an exactly solved model and beyond
• Entanglement renormalization
• Quantum Orders and Symmetric Spin Liquids
• Local stabilizer codes in three dimensions without string logical operators
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Quantum Glassiness
• Fractons
• Fracton Phases of Matter
• A classification of symmetry enriched topological phases with exactly solvable models
• Exotic topological order in fractal spin liquids
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Quantum Field Theory of X-Cube Fracton Topological Order and Robust Degeneracy from Geometry
• Fracton Models on General Three-Dimensional Manifolds
• Fracton topological order via coupled layers
• Fractal symmetries: Ungauging the cubic code
• Foliated fracton order from gauging subsystem symmetries
• Topological Quantum Glassiness
• Quantum order from string-net condensations and origin of light and massless fermions
• Fractional excitations in foliated fracton phases
• Commuting Pauli Hamiltonians as maps between free modules
• X-Cube Fracton Model on Generic Lattices: Phases and Geometric Order
• Twisted Fracton Models in Three Dimensions
• Exotic ${\mathbb{Z}}_N$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Cage-Net Fracton Models
• Symmetric Fracton Matter: Twisted and Enriched
• Bifurcation in entanglement renormalization group flow of a gapped spin model
• Twisted foliated fracton phases
• Fracton phases via exotic higher-form symmetry-breaking
• Universal entanglement signatures of foliated fracton phases
• Fracton fusion and statistics
• Foliated fracton order in the checkerboard model
• Sorting topological stabilizer models in three dimensions
• Bifurcating entanglement-renormalization group flows of fracton stabilizer models
• FCC, Checkerboards, Fractons, and QFT
• Fractonic order in infinite-component Chern-Simons gauge theories
• Fracton Self-Statistics
• Odd Fracton Theories, Proximate Orders, and Parton Constructions
• Emergent fermionic gauge theory and foliated fracton order in the Chamon model
• Homological invariants of Pauli stabilizer codes
• Renormalization of Ising cage-net model and generalized foliation
• Entanglement renormalization of fractonic anisotropic ${\mathbb{Z}}_N$ Laplacian models