Spin Chain with Continuous Exponential Symmetry: Topological Phase and Beyond

arXiv:2610.10682 · cond-mat.str-el, cond-mat.mes-hall, cond-mat.quant-gas, quant-ph · Submitted 2026-10-07 · Read on arXiv

cond-mat.str-el, cond-mat.mes-hall, cond-mat.quant-gas, quant-ph

Submitted: 2026-10-07

Updated: 2026-10-07

Terminology

• Entanglement — 18× in this paper · explained in 3 episodes
• Degeneracy — 16× in this paper · explained in 3 episodes
• Hilbert Space — 10× in this paper · explained in 2 episodes
• Frustration — 6× in this paper · explained in 2 episodes
• Topological Order — 6× in this paper · explained in 2 episodes
• Phase Transitions — 3× in this paper · explained in 2 episodes
• Hubbard Model — 2× in this paper · explained in 3 episodes
• Quantum Circuits — 2× in this paper · explained in 2 episodes
• Symmetry Breaking — 2× in this paper · explained in 3 episodes
• Entanglement Entropy — 1× in this paper · explained in 2 episodes
• Generalization — 1× in this paper · explained in 13 episodes
• Trace — 1× in this paper · explained in 2 episodes

Sources

• Topological order and Fractons from Gauging Exponential Symmetries
• Exponentially slow thermalization in 1D fragmented dynamics
• Translationally Covariant Modulated Symmetries: Classification and Goldstone
• Topological quantum chains protected by dipolar and other modulated symmetries
• Lattice Translation Modulated Symmetries and TFTs
• Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints
• Defect Networks for Topological Phases Protected By Modulated Symmetries
• Matrix Product States for Modulated Symmetries: SPT, LSM, and Beyond
• Projector, Neural, and Tensor-Network Representations of $\mathbb{Z}_N$ Cluster and Dipolar-cluster SPT States

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