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Predicting Entanglement Spectra of Gapless Symmetry-Protected Topological States in One Dimension

Wen-Tao Xu, Frank Pollmann, Michael Knap

PRX QuantumOct 6, 20260 citations

Entanglement Spectra of Stabilizer Codes: A Window into Fracton Order and Sub-System Symmetry Protected Topological States

Albert T. Schmitz, Sheng-Jie Huang, Abhinav Prem

APSJan 1, 20190 citations

A Framework for Predicting Entanglement Spectra of Gapless Symmetry-Protected Topological States in One Dimension

Wen-Tao Xu, Frank Pollmann, Michael Knap

The concept of gapped symmetry-protected topological (SPT) states has been generalized to gapless SPT (gSPT) states. Similar to gapped SPT states, gSPT states in one dimension exhibit universal degeneracies in their entanglement spectra. The entanglement spectra of gSPT states are further described by boundary conformal field theories, whose systematic prediction is a key open question. To address this problem, we focus on the class of gSPT states that are obtained by applying unitary SPT entanglers to trivial, critical states in one dimension. We find that the reduced density matrix of a non-trivial gSPT state can be obtained, either exactly or approximately, by applying a quantum channel to the reduced density matrix of the trivial gSPT state. This quantum channel acts only near the entanglement cut and modifies its corresponding conformal boundary condition, allowing us in turn to predict the boundary conformal field theory describing the entanglement spectra. We apply this framework to gSPT states protected by various symmetries and having different central charges, and further analyze the stability of boundary conditions of the entanglement cut. Our work thereby provides a framework for systematically analyzing and understanding the entanglement spectra of gSPT states.The concept of gapped symmetry-protected topological (SPT) states has been generalized to gapless SPT (gSPT) states. Similar to gapped SPT states, gSPT states in one dimension exhibit universal degeneracies in their entanglement spectra. The entanglement spectra of gSPT states are further described by boundary conformal field theories, whose systematic prediction is a key open question. To address this problem, we focus on the class of gSPT states that are obtained by applying unitary SPT entanglers to trivial, critical states in one dimension. We find that the reduced density matrix of a non-trivial gSPT state can be obtained, either exactly or approximately, by applying a quantum channel to the reduced density matrix of the trivial gSPT state. This quantum channel acts only near the entanglement cut and modifies its corresponding conformal boundary condition, allowing us in turn to predict the boundary conformal field theory describing the entanglement spectra. We apply this framework to gSPT states protected by various symmetries and having different central charges, and further analyze the stability of boundary conditions of the entanglement cut. Our work thereby provides a framework for systematically analyzing and understanding the entanglement spectra of gSPT states.

quant-pharXiv:2604.10128Apr 14, 20260 citations

Entanglement Spectra of Gapped One-dimensional Field Theories and Symmetry-Protected Topological Phases

Gil Young Cho, Ken Shiozaki, Andreas W. W. Ludwig +1 more

Bulletin of the American Physical SocietyMar 15, 20160 citations

Critical entanglement spectrum of one-dimensional symmetry protected topological phases

Guang-Ming Zhang, Xin Wan

Bulletin of the American Physical SocietyMar 2, 20150 citations

Universal Entanglement Spectrum in One-Dimensional Gapless Symmetry Protected Topological States

Xue-Jia Yu, Sheng Yang, Hai‐Qing Lin +1 more

Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.

arXiv:2402.04042Physical Review LettersJul 12, 202439 citations

A Framework for Predicting Entanglement Spectra of Gapless Symmetry-Protected Topological States in One Dimension

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A Framework for Predicting Entanglement Spectra of Gapless Symmetry-Protected Topological States in One Dimension

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