Fractons

Abstract

We review what is known about fracton phases of quantum matter. Fracton phases are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions. They constitute a new class of quantum state of matter, which does not wholly fit into any of the existing paradigms, but which connects to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We introduce the basic phenomena, and discuss the geometric and topological response of fracton phases. We also discuss connections to generalized gauge theories, and explain how gapped fracton phases may be obtained from more familiar theories. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective on fracton phases. We discuss how tensor gauge theory encodes the fracton phenomenon, and how it allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases, and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at non-zero density, before concluding with a brief survey of some open problems.

Review article

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Cited by in corpus (376)

• Localization from Hilbert space shattering: from theory to physical realizations
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• Generalized Symmetries in Condensed Matter
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• Hilbert Space Fragmentation and Commutant Algebras
• 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
• Subdiffusion and heat transport in a tilted 2D Fermi-Hubbard system
• Foliated fracton order from gauging subsystem symmetries
• Towards classification of Fracton phases: the multipole algebra
• Field Theories With a Vector Global Symmetry
• Fractons, dipole symmetries and curved spacetime
• Kinetically constrained freezing transition in a dipole-conserving system
• Measurement as a shortcut to long-range entangled quantum matter
• Anomalous Subdiffusion from Subsystem Symmetries
• Dynamical scar states in driven fracton systems
• Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Long-range entanglement from measuring symmetry-protected topological phases
• Foliated Quantum Field Theory of Fracton Order
• A Modified Villain Formulation of Fractons and Other Exotic Theories
• Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian
• Simple Mitigation of Global Depolarizing Errors in Quantum Simulations
• Hyperbolic Fracton Model, Subsystem Symmetry, and Holography
• Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order
• Multipole conservation laws and subdiffusion in any dimension
• Quantum East model: localization, non-thermal eigenstates and slow dynamics
• The dipolar Bose-Hubbard model
• Fractonic Matter in Symmetry-Enriched U(1) Gauge Theory
• Exotic ${\mathbb{Z}}_N$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Cage-Net Fracton Models
• Fractonic Superfluids
• Rank-2 $U(1)$ spin liquid on the breathing pyrochlore lattice
• Hydrodynamics of ideal fracton fluids
• Crystal-to-Fracton Tensor Gauge Theory Dualities
• Topological Defect Networks for Fractons of all Types
• Spectral statistics in constrained many-body quantum chaotic systems
• Anomalous hydrodynamics in a class of scarred frustration-free Hamiltonians
• Fractons from vector gauge theory
• Spontaneous breaking of multipole symmetries
• Anomaly Inflow for Subsystem Symmetries
• Stark many-body localization: Evidence for Hilbert-space shattering
• An Introduction to Higher-Form Symmetries
• Twisted foliated fracton phases
• Fracton phases via exotic higher-form symmetry-breaking
• More Exotic Field Theories in 3+1 Dimensions
• Fracton fusion and statistics
• Carroll/fracton particles and their correspondence
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Experimental signatures of quantum and topological states in frustrated magnetism
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• Emergence of Hilbert Space Fragmentation in Ising Models with a Weak Transverse Field
• Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry
• Tightening the Lieb-Robinson Bound in Locally-Interacting Systems
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• Scars from protected zero modes and beyond in $U(1)$ quantum link and quantum dimer models
• Flat bands and nontrivial topological properties in an extended Lieb lattice
• Strong planar subsystem symmetry-protected topological phases and their dual fracton orders
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• Spectroscopic fingerprints of gapped quantum spin liquids, both conventional and fractonic
• Hilbert space fragmentation and slow dynamics in particle-conserving quantum East models
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• Hidden quasiconservation laws in fracton hydrodynamics
• Pauli topological subsystem codes from Abelian anyon theories
• Topological Route to New and Unusual Coulomb Spin Liquids
• Fractons with Twisted Boundary Conditions and Their Symmetries
• Self-induced spin glass state in elemental and crystalline neodymium
• Emergent fractons and algebraic quantum liquid from plaquette melting transitions
• Spiral spin-liquid on a honeycomb lattice
• Fractons from confinement in one dimension
• Unifying Emergent Hydrodynamics and Lindbladian Low Energy Spectra across Symmetries, Constraints, and Long-Range Interactions
• Many-body localization in tilted and harmonic potentials
• Position-Dependent Excitations and UV/IR Mixing in the ${\mathbb{Z}}_N$ Rank-2 Toric Code and its Low-Energy Effective Field Theory
• Anyons and Fractional Quantum Hall Effect in Fractal Dimensions
• Gauging permutation symmetries as a route to non-Abelian fractons
• Fragmentation in Frustrated Magnets -- A review
• Logarithmic entanglement growth from disorder-free localization in the two-leg compass ladder
• Comments on Foliated Gauge Theories and Dualities in 3+1d
• Probing localization and quantum geometry by spectroscopy
• Hilbert space shattering and dynamical freezing in the quantum Ising model
• Operator relaxation and the optimal depth of classical shadows
• Fractonic superfluids. (II). Condensing subdimensional particles
• Hohenberg-Mermin-Wagner-type theorems and dipole symmetry
• Rank-2 Toric Code in Two Dimensions
• Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions
• Compactifying fracton stabilizer models
• From Symmetries to Commutant Algebras in Standard Hamiltonians
• $(k,n)$-fractonic Maxwell theory
• Fractons from polarons
• Subsystem symmetry enriched topological order in three dimensions
• Construction of Fractal Order and Phase Transition with Rydberg Atoms
• Foliated fracton order in the Majorana checkerboard model
• 2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• FCC, Checkerboards, Fractons, and QFT
• Dynamics and large deviation transitions of the XOR-Fredrickson-Andersen kinetically constrained model
• Hilbert space shattering and disorder-free localization in polar lattice gases
• Quantum effects on unconventional pinch point singularities
• Probing transport and slow relaxation in the mass-imbalanced Fermi-Hubbard model
• Constructing Quantum Spin Liquids Using Combinatorial Gauge Symmetry
• Kinetically Constrained Quantum Dynamics in Superconducting Circuits
• Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter
• A Neural-Network Variational Quantum Algorithm for Many-Body Dynamics
• Quantum robustness of fracton phases
• Exact solution for the filling-induced thermalization transition in a 1D fracton system
• Quantum phases of two-dimensional ${\mathbb{Z}}_2$ gauge theory coupled to single-component fermion matter
• Spontaneously Broken Subsystem Symmetries
• World-Line Description of Fractons
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• Classifying local fractal subsystem symmetry protected topological phases
• Universality Classes of Stabilizer Code Hamiltonians
• Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems
• Multiflavor Mott insulators in quantum materials and ultracold atoms
• Optimal Thresholds for Fracton Codes and Random Spin Models with Subsystem Symmetry
• Protection of parity-time symmetry in topological many-body systems: non-Hermitian toric code and fracton models
• Maxwell theory of fractons
• Foliated BF theories and Multipole symmetries
• Subdimensional criticality: condensation of lineons and planons in the X-cube model
• Strong zero modes in a class of generalised Ising spin ladders with plaquette interactions
• Effective field theories for gapless phases with fractons via a coset construction
• Hyperbolic Fracton Model, Subsystem Symmetry, and Holography II: The Dual Eight-Vertex Model
• Fractonic order in infinite-component Chern-Simons gauge theories
• Low energy structure of spiral spin liquids
• Coupled Hydrodynamics in Dipole-Conserving Quantum Systems
• Hilbert space fragmentation produces a "fracton Casimir effect"
• Interaction Enabled Fractonic Higher-Order Topological Phases
• The theory of symmetric tensor field: from fractons to gravitons and back
• Fractons and exotic symmetries from branes
• Boson-fermion duality with subsystem symmetry
• Two-dimensional melting via sine-Gordon duality
• Gapless edge modes in (4+1)-dimensional topologically massive tensor gauge theory and anomaly inflow for subsystem symmetry
• Dual gauge theory formulation of planar quasicrystal elasticity and fractons
• Gauging fractons and linearized gravity
• Topological fracton quantum phase transitions by tuning exact tensor network states
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• Higher-Rank Tensor Non-Abelian Field Theory: Higher-Moment or Subdimensional Polynomial Global Symmetry, Algebraic Variety, Noether's Theorem, and Gauging
• Emergent tracer dynamics in constrained quantum systems
• Fractional Chiral Hinge Insulator
• Critically slow operator dynamics in constrained many-body systems
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Fracton Topological Order at Finite Temperature
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Boundary theory of the X-cube model in the continuum
• A Chern-Simons theory for dipole symmetry
• Classical Dimers on Penrose Tilings
• Foliated-Exotic Duality in Fractonic BF Theories
• A symmetry principle for gauge theories with fractons
• Height-conserving quantum dimer models
• Ideal fracton superfluids
• Local Integrals of Motion for Topologically Ordered Many-Body Localized Systems
• Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements
• Planar p-String Condensation: Chiral Fracton Phases from Fractional Quantum Hall Layers and Beyond
• Spontaneously broken supersymmetric fracton phases with fermionic subsystem symmetries
• Effective Field Theory of Dipolar Braiding Statistics in Two Dimensions
• Covariant fracton gauge theory with boundary
• Fracton magnetohydrodynamics
• Gapped Lineon and Fracton Models on Graphs
• Fracton physics of spatially extended excitations. II. Polynomial ground state degeneracy of exactly solvable models
• A degeneracy bound for homogeneous topological order
• Fractonic gauge theory of smectics
• On the stability of topological order in tensor network states
• Bifurcating subsystem symmetric entanglement renormalization in two dimensions
• Geometric Criterion for Solvability of Lattice Spin Systems
• Lifshitz symmetry: Lie algebras, spacetimes and particles
• Non-Abelian Gauged Fracton Matter Field Theory: New Sigma Models, Superfluids and Vortices
• Deconfinement Dynamics of Fractons in Tilted Bose-Hubbard Chains
• Gapped boundaries of (3+1)d topological orders
• Fracton-elasticity duality in twisted moiré superlattices
• Fracton Self-Statistics
• Fractonic Higher-Order Topological Phases in Open Quantum Systems
• Emergent Fracton Dynamics in a Non-Planar Dimer Model
• Exotic quantum liquids in Bose-Hubbard models with spatially-modulated symmetries
• Flux mobility delocalization in the Kitaev spin ladder
• Lattice control of non-ergodicity in a polar lattice gas
• Higher-Rank Tensor Field Theory of Non-Abelian Fracton and Embeddon
• Hierarchical hydrodynamics in long-range multipole-conserving systems
• Carrollian Yang-Mills Theory
• Lattice Clifford fractons and their Chern-Simons-like theory
• Competing topological orders in three dimensions: X-cube versus toric code
• Evolution of Dynamical Signature in the X-cube Fracton Topological Order
• Reduced Conformal Symmetry
• Fractons, non-Riemannian Geometry, and Double Field Theory
• Fractons on Graphs and Complexity
• Colloquium: Synthetic quantum matter in non-standard geometries
• Fracton gravity from spacetime dipole symmetry
• Symmetries as Ground States of Local Superoperators: Hydrodynamic Implications
• Detecting Subsystem Symmetry Protected Topological Order Through Strange Correlators
• A cellular automaton decoder for a noise-bias tailored color code
• Realization of fractonic quantum phases in the breathing pyrochlore lattice
• Renormalization group analysis on emergence of higher rank symmetry and higher moment conservation
• Symmetry protected self correcting quantum memory in three space dimensions
• Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings
• Tearing Fractons
• Fractons from frustration in hole-doped antiferromagnets
• Non-Liquid Cellular States
• Lieb Schultz Mattis-Type Theorems and Other Non-perturbative Results for Strongly Correlated Systems with Conserved Dipole Moments
• Hilbert Space Fragmentation and Ashkin-Teller Criticality in Fluctuation Coupled Ising Models
• Classification and construction of interacting fractonic higher-order topological phases
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• Slow dynamics of a mobile impurity interacting with an Anderson insulator
• Pascal's Triangle Fractal Symmetries
• Noncommutative Field Theory of the Tkachenko Mode: Symmetries and Decay Rate
• Topological Dipole Conserving Insulators and Multipolar Responses
• Multipole groups and fracton phenomena on arbitrary crystalline lattices
• Scalar, fermionic and supersymmetric field theories with subsystem symmetries in d+1 dimensions
• Designer non-Abelian fractons from topological layers
• A Classical Chiral Spin-Liquid from Chiral Interactions on the Pyrochlore Lattice
• Fractonic topological phases from coupled wires
• An integrable spin chain with Hilbert space fragmentation and solvable real time dynamics
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Fractons from a liquid of singlet pairs
• Emergent dipole moment conservation and subdiffusion in tilted chains
• Fractalizing quantum codes
• Beyond the Freshman's Dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models
• Exact pre-transition effects in kinetically constrained circuits: dynamical fluctuations in the Floquet-East model
• Odd Fracton Theories, Proximate Orders, and Parton Constructions
• Freezing transition in particle-conserving East model
• Boundary conditions dependence of the phase transition in the quantum Newman-Moore model
• Electric Circuit Realizations of Fracton Physics
• Spectroscopic fingerprints of gapless type-II fracton phases
• Fracton-like phases from subsystem symmetries
• Boundary Modes in the Chamon Model
• Fracton excitations in classical frustrated kagome spin models
• Holographic subdiffusion
• Many-body Localization in Clean Chains with Long-Range Interactions
• Classical Non-Relativistic Fractons
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Non-Fermi Liquids from Dipolar Symmetry Breaking
• Comments on Quivers and Fractons
• Reweight-annealing method for evaluating the partition function via quantum Monte Carlo calculations
• Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems
• Screw dislocations in the X-cube fracton model
• Non-Fermi liquid induced by Bose metal with protected subsystem symmetries
• Quantum vortex lattice: Lifshitz duality, topological defects and multipole symmetries
• Disentangling (2+1)d Topological States of Matter with the Entanglement Negativity
• Quantum many-body scars with unconventional superconducting pairing symmetries via multibody interactions
• Dynamical Spectral Response of Fractonic Quantum Matter
• Hierarchy of Entanglement Renormalization and Long-Range Entangled States
• Non-relativistic spin-3 symmetries in 2+1 dimensions from expanded/extended Nappi-Witten algebras
• Asymptotic symmetries and soft charges of fractons
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• Exotic Symmetry Breaking Properties of Self-Dual Fracton Spin Models
• Crystal gravity
• Entanglement Signature of Hinge Arcs, Fermi Arcs, and Crystalline Symmetry Protection in Higher-Order Weyl Semimetals
• Covariant Interacting Fractons
• Line-Graph Approach to Spiral Spin Liquids
• Spiral spin liquid in a frustrated honeycomb antiferromagnet: A single-crystal study of GdZnPO
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• Quantum Hydrodynamics of Fractonic Superfluids with Lineon Condensate: from Navier-Stokes-like Equations to Landau-like Criterion
• Topic Review: Hatsugai-Kohmoto models: Exactly solvable playground for Mottness and Non-Fermi Liquid
• Linked Cluster Expansions via Hypergraph Decompositions
• Exploring Bosonic and Fermionic Link Models on $(3+1)-$d tubes
• Fractons in effective field theories for spontaneously broken translations
• Generalized toric codes on twisted tori for quantum error correction
• Non-local finite-depth circuits for constructing SPT states and quantum cellular automata
• Renormalization of Ising cage-net model and generalized foliation
• Fractal Subsystem Symmetries, 't Hooft Anomalies, and UV/IR Mixing
• Hamiltonian Formulation of Higher Rank Symmetric Gauge Theories
• Quasi-topological fractons: a 3D dipolar gauge theory
• Observation of the spiral spin liquid in a triangular-lattice material
• Deformed Fredkin model for the $\nu =5/2$ Moore-Read state on thin cylinders
• Towards absolutely stable ergodicity breaking in two and three dimensions
• Irrational moments and signatures of higher-rank gauge theories in diluted classical spin liquids
• Local integrals of motion in dipole-conserving models with Hilbert space fragmentation
• Interacting Fractons in 2+1-Dimensional Quantum Field Theory
• [111]-strained spin ice: Localization of thermodynamically deconfined monopoles
• $\mathcal{N}=1$ Curves on Generalized Coulomb Branches
• A post-Gaussian approach to dipole symmetries and interacting fractons
• Residual entropy from temperature incremental Monte Carlo method
• Quantum effects on pyrochlore higher-rank U(1) spin liquids: Pinch-line singularities, spin nematics, and connections to oxide materials
• Foliated order parameter in a fracton phase transition
• Fractal Subsystem Symmetries, Anomalies, Boundaries, and Effective Field Theory
• Three-dimensional fracton topological orders with boundary Toeplitz braiding
• Chiral Virasoro algebra from a single wavefunction
• A New Model for Fractons, Fluxons, and Freeons
• Self-correction from higher-form symmetry protection on a boundary
• Fracton and Non-Lorentzian Particle Duality: Gauge Field Couplings and Geometric Implications
• Corner Modes and Ground-State Degeneracy in Models with Gauge-Like Subsystem Symmetries
• Critical region of topological trivial and nontrivial phases in interacting Kitaev chain with spatially varying potentials
• Human-machine collaboration: ordering mechanism of rank-2 spin liquid on breathing pyrochlore lattice
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Tracking locality in time evolution of disordered systems
• Fractons in curved space
• Machian fractons, Hamiltonian attractors and non-equilibrium steady states
• Classical Fractons: Local chaos, global broken ergodicity and an arrow of time
• The spin-orbital Kitaev model: from kagome spin ice to classical fractons
• Fracton and topological order in the XY checkerboard toric code
• Landscapes of an out-of-equilibrium anyonic sea
• Strong Hilbert space fragmentation and fractons from subsystem and higher-form symmetries
• Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum
• Non-equilibrium dynamics of bosons with dipole symmetry: Large-$N$ Keldysh approach
• A Multicritical Point with Infinite Fractal Symmetries
• Neural Quantum State Study of Fracton Models
• Scaling limit of the staggered six-vertex model with $U_q(\mathfrak{sl}(2))$ invariant boundary conditions
• Bosonic Quantum Breakdown Hubbard Model
• Advancing Hybrid Quantum-Classical Algorithms via Mean-Operators
• Floating topological phases
• Non-Abelian fracton order from gauging a mixture of subsystem and global symmetries
• Higgs Phases and Boundary Criticality
• Fractonic criticality in Rydberg atom arrays
• Strings as Hyper-Fractons
• Two Channel Kondo behavior in the quantum XX chain with a boundary defect
• String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to $p$-String Condensation
• Frustration and chirality in three-dimensional trillium lattices: Insights and Perspectives
• A Theorem on Extensive Spectral Degeneracy for Systems with Higher Symmetries in General Dimensions
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• Dual Description of Gauge Theories from an Iterative Noether Approach
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• The Symmetry Principle in Condensed Matter Physics (I)
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• Partially topological phase in a quantum loop gas model with tension and pressure
• Topological Dipole Insulator
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• Classical fracton spin liquid and Hilbert space fragmentation in a 2D spin-$1/2$ model
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• Planons and their Carroll-Galilei symmetries
• Covariant field theory of 3D massive fractons
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• (Four) Dual Plaquette 3D Ising Models
• Fractonic superfluids. III. Hybridizing higher moments
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• Galilei particles revisited
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• Phase space fractons
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• Flavored extended instanton in QCD
• ${\mathbb{Z}}_N$ generalizations of three-dimensional stabilizer codes
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• A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules
• Resonant dynamics of dipole-conserving Bose-Hubbard model with time-dependent tensor electric fields
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• There and Back Again: A Gauging Nexus between Topological and Fracton Phases
• Quantum Fractality on the Surface of Topological Insulators
• Infinite-component $BF$ field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
• Itinerant and topological excitations in a honeycomb spiral spin liquid candidate
• Dynamics of quasiholes and quasiparticles at the edges of small lattices
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• Entanglement renormalization of fractonic anisotropic ${\mathbb{Z}}_N$ Laplacian models
• Continuum Fractons: Quantization and the Few Body Problem
• Non-Commutative weak measurements: Entanglement, Symmetry Breaking, and the Role of Readout
• Mapping the Phase Diagram of a Frustrated Magnet: Degeneracies, Flat Bands, and Canting Cycles on the Pyrochlore Lattice
• Symmetry-enriched topological order and quasifractonic behavior in ${\mathbb{Z}}_N$ stabilizer codes
• Immobile and mobile excitations of three-spin interactions on the diamond chain
• Hollow Lattice Tensor Gauge Theories with Bosonic Matter
• Error Resilience of Fracton Codes and Near Saturation of Code-Capacity Threshold in Three Dimensions
• Emergent fracton strings from covariant bi-form gauge field theory
• Exotic theta terms in 2+1d fractonic field theory
• Covariant Fracton Electrodynamics in Six Dimensions
• Quadrupole-conserving dynamics in the non-commutative plane
• Planon-modular fracton orders
• A non-lorentzian primer
• Emergent criticality in a constrained boson model
• Emergent dipole field theory in atomic ladders
• Constrained motions and slow dynamics in one-dimensional bosons with double-well dispersion
• Localization and melting of interfaces in the two-dimensional quantum Ising model
• Tri-unitary quantum circuits
• Fracton models from product codes
• Structures of group-15 elemental solids from an effective boundary theory
• Subsystem symmetries, critical Bose surface, and immobile excitations in an extended compass model
• Defects in Wigner crystals: fracton-elasticity duality and vacancy proliferation
• Flat bands and band touching from real-space topology in hyperbolic lattices
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• Matrix product state classification of 1D multipole symmetry protected topological phases
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• Collective dynamics in holographic fractonic solids