The equivalence of topological indices for U(1) Z 2-symmetric quantum spin chains

arXiv:2609.34119 · math-ph, cond-mat.stat-mech, cond-mat.str-el, math.MP, quant-ph · Submitted 2026-09-28 · Read on arXiv

math-ph, cond-mat.stat-mech, cond-mat.str-el, math.MP, quant-ph

Submitted: 2026-09-28

Updated: 2026-09-28

Terminology

• Entanglement — 4× in this paper · explained in 3 episodes
• Symmetry Breaking — 4× in this paper · explained in 3 episodes
• Quantization — 2× in this paper · explained in 12 episodes
• Topological Order — 2× in this paper · explained in 2 episodes
• Entanglement Entropy — 1× in this paper · explained in 2 episodes
• Hilbert Space — 1× in this paper · explained in 2 episodes

Sources

• Ground State Properties of Antiferromagnetic Chains with Unrestricted Spin: Integer Spin Chains as Realisations of the O(3) Non-Linear Sigma Model
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protection of topological order in one-dimensional quantum spin systems
• String order and symmetries in quantum spin lattices
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Classifying quantum phases using Matrix Product States and PEPS
• On gapped phases with a continuous symmetry and boundary operators
• Topological phase transition and $\mathbb{Z}_2$ index for $S=1$ quantum spin chains
• A ${\mathbb Z}_2$-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains
• Boundedness of Entanglement Entropy,and Split Property of Quantum Spin Chains
• An Area Law for One Dimensional Quantum Systems
• Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
• Automorphic equivalence within gapped phases in the bulk
• Order Parameter to Characterize Valence-Bond-Solid States in Quantum Spin Chains
• Lieb-Schultz-Mattis theorem with a local twist for general one-dimensional quantum systems
• Rigorous Index Theory for One-Dimensional Interacting Topological Insulators
• Provably efficient machine learning for quantum many-body problems
• The Ground State of the S=1 Antiferromagnetic Heisenberg Chain is Topologically Nontrivial if Gapped
• Quantum spin systems on infinite lattices

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