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arXiv:2604.10128

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

Wen-Tao Xu, Frank Pollmann, Michael Knap

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Apr 11, 2026• Announced Apr 14, 2026• 0 citations
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Abstract

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.

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Main text contains 12 pages, and 10 figures

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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.
Abstract Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking ‘Kitaev chain’ superconductor (symmetry class D), the Haldane phase, and the Z 8 classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.
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.
Symmetry topological field theory (SymTFT) gives a holographic correspondence between systems with a global symmetry and a higher-dimensional topological field theory. In this framework, classification of gapped phases of matter in spacetime dimension 1+1D correspond to classifications of mechanisms to confine the SymTFT by condensing anyons. In this work, we extend these results to characterize gapless symmetry-protected topological states: symmetry-enriched gapless phases or critical points that exhibit edge modes protected by symmetry and topology. We establish a one-to-one correspondence between 1+1D bosonic gSPTs, and partially-confined boundaries of 2+1D SymTFTs. From general physical considerations, we determine the set of data and consistency conditions required to define a 1+1D gSPT, and show that this data precisely matches that of symmetry-preserving partial confinement (or partially gapped boundaries) of 2+1D quantum double models. We illustrate this correspondence through a dimensional reduction (thin-slab) construction, which enables a physically-intuitive derivation of how properties of the gSPT such as edge modes, emergent anomalies, and stability to perturbations arise from the SymTFT perspective.ditions required to define a 1+1D gSPT and show that they fully determine the physics of the gSPT including edge modes and emergent anomaly.
We systematically study gapless edge modes corresponding to Z(3) symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge c=1, given by the coset SUk(3)/SUk(2) CFT at level k=1.
Symmetry-protected topological phases (SPTs) have universal degeneracies in the entanglement spectrum in one dimension. Here we formulate this phenomenon in the framework of symmetry-resolved entanglement (SRE) using cohomology theory. We develop a general approach to compute entanglement measures of SPTs in any dimension and specifically SRE via a discrete path integral on multisheet Riemann surfaces with generalized defects. The resulting path integral is expressed in terms of group cocycles describing the topological actions of SPTs. Their cohomology classification allows us to identify universal entanglement properties. Specifically, we demonstrate an equiblock decomposition of the reduced density matrix into symmetry sectors, for all one-dimensional topological phases protected by finite Abelian unitary symmetries.
We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group GGG in 2+12+12+1 dimensions. Our method is based on directly studying the 1+11+11+1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the in-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry GGG. We discuss the relation between edgeability, i.e., the ability to find a consistent boundary state, and gappability, i.e., the ability to gap out a CFT, in the presence of GGG. We study several cases including time-reversal symmetric topological insulators, ZN{\mathbb{Z}}_{N}ZN​ symmetric bosonic SPT phases, and ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetric topological superconductors.