SciPost Physics

SciPost Phys. 6, 007 (2019) · published 16 January 2019

• doi: 10.21468/SciPostPhys.6.1.007
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Abstract

We study spin systems which exhibit symmetries that act on a fractal subset of sites, with fractal structures generated by linear cellular automata. In addition to the trivial symmetric paramagnet and spontaneously symmetry broken phases, we construct additional fractal symmetry protected topological (FSPT) phases via a decorated defect approach. Such phases have edges along which fractal symmetries are realized projectively, leading to a symmetry protected degeneracy along the edge. Isolated excitations above the ground state are symmetry protected fractons, which cannot be moved without breaking the symmetry. In 3D, our construction leads additionally to FSPT phases protected by higher form fractal symmetries and fracton topologically ordered phases enriched by the additional fractal symmetries.

• Devakul et al., Fractalizing quantum codes
  Quantum 5, 438 438 (2021) [Crossref]
• Slagle et al., Symmetric tensor gauge theories on curved spaces
  Annals of Physics 410, 167910 167910 (2019) [Crossref]
• Herringer et al., Measurement-Based Quantum Computation in Symmetry-Enriched Topological Phases
  PRX Quantum 6, 040314 (2025) [Crossref]
• Sfairopoulos et al., Boundary conditions dependence of the phase transition in the quantum Newman-Moore model
  Phys. Rev. B 108, 174107 (2023) [Crossref]
• Iliasov et al., Hall conductivity of a Sierpiński carpet
  Phys. Rev. B 101, 045413 (2020) [Crossref]
• Causer et al., Rejection-free quantum Monte Carlo in continuous time from transition path sampling
  Phys. Rev. B 109, 024307 (2024) [Crossref]
• Kim et al., Noninvertible symmetry and topological holography for modulated SPT in one dimension
  SciPost Phys. 19, 110 (2025) [Crossref]
• Stephen et al., Subsystem symmetry enriched topological order in three dimensions
  Phys. Rev. Research 2, 033331 (2020) [Crossref]
• Kumar et al., Symmetry-enforced fractonicity and two-dimensional quantum crystal melting
  Phys. Rev. B 100, 045119 (2019) [Crossref]
• Hsin et al., Subsystem symmetry fractionalization and foliated field theory
  SciPost Phys. 18, 147 (2025) [Crossref]
• Doshi et al., Vortices as fractons
  Commun Phys 4, 44 (2021) [Crossref]
• Pancotti et al., Quantum East Model: Localization, Nonthermal Eigenstates, and Slow Dynamics
  Phys. Rev. X 10, 021051 (2020) [Crossref]
• Tantivasadakarn, Jordan-Wigner dualities for translation-invariant Hamiltonians in any dimension: Emergent fermions in fracton topological order
  Phys. Rev. Research 2, 023353 (2020) [Crossref]
• Osborne et al., Infinite families of fracton fluids with momentum conservation
  Phys. Rev. B 105, 024311 (2022) [Crossref]
• Tantivasadakarn et al., Searching for fracton orders via symmetry defect condensation
  Phys. Rev. B 101, 165143 (2020) [Crossref]
• Tantivasadakarn et al., Hybrid fracton phases: Parent orders for liquid and nonliquid quantum phases
  Phys. Rev. B 103, 245136 (2021) [Crossref]
• Pai et al., Localization in Fractonic Random Circuits
  Phys. Rev. X 9, 021003 (2019) [Crossref]
• Pretko et al., Crystal-to-fracton tensor gauge theory dualities
  Phys. Rev. B 100, 134113 (2019) [Crossref]
• Duan et al., Review about the Application of Fractal Theory in the Research of Packaging Materials
  Materials 14, 860 (2021) [Crossref]
• You et al., Building fracton phases by Majorana manipulation
  Phys. Rev. Research 1, 013011 (2019) [Crossref]
• You et al., Fractonic Chern-Simons and BF theories
  Phys. Rev. Research 2, 023249 (2020) [Crossref]
• Sous et al., Fractons from frustration in hole-doped antiferromagnets
  npj Quantum Mater. 5, 81 (2020) [Crossref]
• Jia et al., Subsystem symmetry-protected topological phases from subsystem SymTFT of 2-foliated exotic tensor gauge theory
  J. High Energ. Phys. 2025, 170 (2025) [Crossref]
• Burnell et al., Anomaly inflow for subsystem symmetries
  Phys. Rev. B 106, 085113 (2022) [Crossref]
• Inack et al., Neural Annealing and Visualization of Autoregressive Neural Networks in the Newman–Moore Model
  Condensed Matter 7, 38 (2022) [Crossref]
• Lehmann et al., Fragmentation-induced localization and boundary charges in dimensions two and above
  SciPost Phys. 14, 140 (2023) [Crossref]
• Dubinkin et al., Theory of dipole insulators
  Phys. Rev. B 103, 125129 (2021) [Crossref]
• dos Anjos et al., Fractality in resistive circuits: the Fibonacci resistor networks
  Eur. Phys. J. B 97, 121 (2024) [Crossref]
• Stephen et al., Detecting subsystem symmetry protected topological order via entanglement entropy
  Phys. Rev. B 100, 115112 (2019) [Crossref]
• Gromov et al., Fracton hydrodynamics
  Phys. Rev. Research 2, 033124 (2020) [Crossref]
• Pace et al., Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
  SciPost Phys. 18, 021 (2025) [Crossref]
• Dua et al., Compactifying fracton stabilizer models
  Phys. Rev. B 99, 245135 (2019) [Crossref]
• Stephen et al., Nonlocal Finite-Depth Circuits for Constructing Symmetry-Protected Topological States and Quantum Cellular Automata
  PRX Quantum 5, 010304 (2024) [Crossref]
• Sun et al., Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems
  PRX Quantum 6, 020333 (2025) [Crossref]
• Prem et al., Cage-Net Fracton Models
  Phys. Rev. X 9, 021010 (2019) [Crossref]
• Sala et al., Dynamics in Systems with Modulated Symmetries
  Phys. Rev. Lett. 129, 170601 (2022) [Crossref]
• Liu et al., Plaquette models, cellular automata, and measurement-induced criticality
  J. Phys. A: Math. Theor. 57, 435003 (2024) [Crossref]
• May-Mann et al., Interaction-enabled fractonic higher-order topological phases
  Phys. Rev. B 105, 245122 (2022) [Crossref]
• Cao et al., Generating lattice non-invertible symmetries
  SciPost Phys. 17, 104 (2024) [Crossref]
• Devakul et al., Strong planar subsystem symmetry-protected topological phases and their dual fracton orders
  Phys. Rev. Research 2, 012059 (2020) [Crossref]
• Stephen et al., Fractionalization of subsystem symmetries in two dimensions
  Phys. Rev. B 106, 085104 (2022) [Crossref]
• dos Anjos et al., The Fractal Geometry of Growth: Fluctuation–Dissipation Theorem and Hidden Symmetry
  Front. Phys. 9, 741590 (2021) [Crossref]
• Gomes-Filho et al., The fluctuation–dissipation relations: Growth, diffusion, and beyond
  Physics Reports 1141, 1 (2025) [Crossref]
• You et al., Higher-order topological superconductors as generators of quantum codes
  Phys. Rev. B 100, 054513 (2019) [Crossref]
• Williamson et al., Spurious Topological Entanglement Entropy from Subsystem Symmetries
  Phys. Rev. Lett. 122, 140506 (2019) [Crossref]
• Dyakin-Sosnovsky, Fractal Biology — Evolution from Molecular to Cognitive, and Psychological Dimensions
  Qeios 6, X0DUH1.2 (2024) [Crossref]
• Cuiper et al., Systematic construction of stabilizer codes via gauging abelian boundary symmetries
  Quantum 9, 1852 1852 (2025) [Crossref]
• Vasiloiu et al., Trajectory phase transitions in noninteracting spin systems
  Phys. Rev. E 101, 042115 (2020) [Crossref]
• Shirley et al., Twisted foliated fracton phases
  Phys. Rev. B 102, 115103 (2020) [Crossref]
• Wang et al., Fractonic superfluids. III. Hybridizing higher moments
  Phys. Rev. Research 7, 033118 (2025) [Crossref]
• Azevedo et al., Fibonacci Resistors Networks
  SSRN Journal, (2022) [Crossref]
• Slagle et al., Foliated field theory and string-membrane-net condensation picture of fracton order
  SciPost Phys. 6, 043 (2019) [Crossref]
• Yan, Hyperbolic fracton model, subsystem symmetry, and holography. II. The dual eight-vertex model
  Phys. Rev. B 100, 245138 (2019) [Crossref]
• Stephen et al., Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
  Quantum 3, 142 142 (2019) [Crossref]
• Zhou et al., Detecting subsystem symmetry protected topological order through strange correlators
  Phys. Rev. B 106, 214428 (2022) [Crossref]
• Zhang et al., Fractonic higher-order topological phases in open quantum systems
  Phys. Rev. B 108, 155123 (2023) [Crossref]
• Noh et al., Construction of entangled many-body states via the Higgs mechanism
  Phys. Rev. B 107, 054401 (2023) [Crossref]
• Biswas et al., Beyond the freshman's dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models
  Phys. Rev. B 105, 224410 (2022) [Crossref]
• Fechisin et al., Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State
  Phys. Rev. X 15, 011058 (2025) [Crossref]
• Kibe et al., Stabilizer code model with noninvertible symmetries: Strange fractons, confinement, and noncommutative and non-Abelian fusion rules
  Phys. Rev. D 110, 105019 (2024) [Crossref]
• de Lima et al., Scaling, fractal dynamics, and critical exponents: Application in a noninteger-dimensional Ising model
  Phys. Rev. E 112, 044109 (2025) [Crossref]
• Dua et al., Bifurcating entanglement-renormalization group flows of fracton stabilizer models
  Phys. Rev. Research 2, 033021 (2020) [Crossref]
• Nixon et al., Correcting Spanning Errors With a Fractal Code
  IEEE Trans. Inform. Theory 67, 4504 (2021) [Crossref]
• Casasola et al., Fractal subsystem symmetries, anomalies, boundaries, and effective field theory
  Phys. Rev. B 110, 195105 (2024) [Crossref]
• Sfairopoulos et al., Quantum Newman-Moore model in a longitudinal field
  Phys. Rev. B 112, 094103 (2025) [Crossref]
• Maier et al., Topological Order in Symmetric Blockade Structures
  PRX Quantum 6, 030340 (2025) [Crossref]
• Williamson et al., Type-II fractons from coupled spin chains and layers
  Phys. Rev. B 103, 155140 (2021) [Crossref]
• Tantivasadakarn et al., Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
  Phys. Rev. Lett. 129, 090501 (2022) [Crossref]
• Prem et al., Gauging permutation symmetries as a route to non-Abelian fractons
  SciPost Phys. 7, 068 (2019) [Crossref]
• You, Non-Abelian defects in fracton phases of matter
  Phys. Rev. B 100, 075148 (2019) [Crossref]
• Zhang et al., Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi Point Strange Correlators
  PRX Quantum 5, 030342 (2024) [Crossref]
• He et al., Lieb-Schultz-Mattis–type constraints on fractonic matter
  Phys. Rev. B 101, 165145 (2020) [Crossref]
• Cao et al., Boson-fermion duality with subsystem symmetry
  Phys. Rev. B 106, 075150 (2022) [Crossref]
• Lima et al., Geometrical interpretation of critical exponents
  Phys. Rev. E 110, L062107 (2024) [Crossref]
• Sous et al., Fractons from polarons
  Phys. Rev. B 102, 214437 (2020) [Crossref]
• Zhang et al., Classification and construction of interacting fractonic higher-order topological phases
  Phys. Rev. B 108, 045133 (2023) [Crossref]
• Devakul, Classifying local fractal subsystem symmetry-protected topological phases
  Phys. Rev. B 99, 235131 (2019) [Crossref]
• Casasola et al., Fractal subsystem symmetries, 't Hooft anomalies, and UV/IR mixing
  Phys. Rev. B 109, 075164 (2024) [Crossref]
• Shirley et al., Foliated fracton order in the checkerboard model
  Phys. Rev. B 99, 115123 (2019) [Crossref]
• Shirley et al., Foliated fracton order from gauging subsystem symmetries
  SciPost Phys. 6, 041 (2019) [Crossref]
• Miguel et al., A cellular automaton decoder for a noise-bias tailored color code
  Quantum 7, 940 940 (2023) [Crossref]
• Lam, Classification of dipolar symmetry-protected topological phases: Matrix product states, stabilizer Hamiltonians, and finite tensor gauge theories
  Phys. Rev. B 109, 115142 (2024) [Crossref]
• San Miguel et al., Bifurcating subsystem symmetric entanglement renormalization in two dimensions
  Phys. Rev. B 103, 035148 (2021) [Crossref]
• Song et al., Twisted fracton models in three dimensions
  Phys. Rev. B 99, 155118 (2019) [Crossref]
• Schmitz et al., Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter
  Phys. Rev. B 99, 205109 (2019) [Crossref]
• Pace, Emergent generalized symmetries in ordered phases and applications to quantum disordering
  SciPost Phys. 17, 080 (2024) [Crossref]
• You et al., Multipolar topological field theories: Bridging higher order topological insulators and fractons
  Phys. Rev. B 103, 245128 (2021) [Crossref]
• You, Quantum Liquids: Emergent Higher-Rank Gauge Theory and Fractons
  16, 83 (2025) [Crossref]
• McGreevy, Generalized Symmetries in Condensed Matter
  Annu. Rev. Condens. Matter Phys. 14, 57 (2023) [Crossref]
• Gromov, Towards Classification of Fracton Phases: The Multipole Algebra
  Phys. Rev. X 9, 031035 (2019) [Crossref]
• Canossa et al., Exotic symmetry breaking properties of self-dual fracton spin models
  Phys. Rev. Research 6, 013304 (2024) [Crossref]
• Han et al., Topological quantum chains protected by dipolar and other modulated symmetries
  Phys. Rev. B 109, 125121 (2024) [Crossref]
• Myerson-Jain et al., Multicritical point with infinite fractal symmetries
  Phys. Rev. B 106, 115130 (2022) [Crossref]
• Williamson et al., Fractonic matter in symmetry-enriched U(1) gauge theory
  Phys. Rev. B 100, 125150 (2019) [Crossref]
• Myerson-Jain et al., Construction of Fractal Order and Phase Transition with Rydberg Atoms
  Phys. Rev. Lett. 128, 017601 (2022) [Crossref]