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    "title": "Symmetry-protected entanglement degeneracy characterizes the Haldane phase",
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        "description": "The derivation uses a canonical pure matrix-product-state description with a unique leading transfer eigenvalue and the nontrivial symmetry sector. Preserving a symmetry alone does not force every state into the Haldane phase. Inversion protection refers to an inversion-symmetric cut; ordinary translation symmetry is not the essential protection. Bond weakening preserves an antisymmetric state sector and need not end in the absolute ground state of the disconnected halves. The numerical demonstrations use finite bond dimension, and the general classification of all one-dimensional gapped phases is deferred in this source.",
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      "statement": "# Symmetry-protected entanglement degeneracy characterizes the Haldane phase\n\nPollmann, Turner, Berg and Oshikawa characterized the Haldane phase through an even degeneracy of its entire entanglement spectrum. Their matrix-product-state analysis relates this degeneracy to how symmetry acts on the Schmidt states. Bond-centered inversion, time reversal or the dihedral group of spin rotations can each protect the nontrivial structure. This identifies the phase even when conventional string order or zero-energy physical edge states are absent. Numerical spin-chain examples and a symmetry-preserving bond-weakening process illustrate the distinction from a trivial phase.\n\n## Scientific scope\n\nThe derivation uses a canonical pure matrix-product-state description with a unique leading transfer eigenvalue and the nontrivial symmetry sector. Preserving a symmetry alone does not force every state into the Haldane phase. Inversion protection refers to an inversion-symmetric cut; ordinary translation symmetry is not the essential protection. Bond weakening preserves an antisymmetric state sector and need not end in the absolute ground state of the disconnected halves. The numerical demonstrations use finite bond dimension, and the general classification of all one-dimensional gapped phases is deferred in this source.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nFrank Pollmann; Ari M. Turner; Erez Berg; Masaki Oshikawa\n\nThese are the listed authors of the cited contribution sources. Paper authorship does not quantify individual effort. An incomplete list is explicitly marked; same names are not automatically merged.\n\n\n\n## Original sources\n\n- [Entanglement spectrum of a topological phase in one dimension (arXiv0910.1811v2)](https://arxiv.org/pdf/0910.1811v2)\n",
      "summary": "Pollmann, Turner, Berg and Oshikawa characterized the Haldane phase through an even degeneracy of its entire entanglement spectrum. Their matrix-product-state analysis relates this degeneracy to how symmetry acts on the Schmidt states. Bond-centered inversion, time reversal or the dihedral group of spin rotations can each protect the nontrivial structure. This identifies the phase even when conventional string order or zero-energy physical edge states are absent. Numerical spin-chain examples and a symmetry-preserving bond-weakening process illustrate the distinction from a trivial phase.",
      "title": "Symmetry-protected entanglement degeneracy characterizes the Haldane phase"
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    "what_changed": "Pollmann, Turner, Berg and Oshikawa characterized the Haldane phase through an even degeneracy of its entire entanglement spectrum. Their matrix-product-state analysis relates this degeneracy to how symmetry acts on the Schmidt states. Bond-centered inversion, time reversal or the dihedral group of spin rotations can each protect the nontrivial structure. This identifies the phase even when conventional string order or zero-energy physical edge states are absent. Numerical spin-chain examples and a symmetry-preserving bond-weakening process illustrate the distinction from a trivial phase.",
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