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.
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.
Topological phenomena in quantum critical systems have recently attracted growing attention, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, a general framework for identifying such nontrivial phenomena, particularly in higher-dimensional systems, remains insufficiently explored. In this work, we propose a universal fingerprint for detecting nontrivial topology in critical free-fermion systems protected by global on-site symmetries. Specifically, we analytically establish an exact relation between the bulk entanglement spectrum and the boundary energy spectrum at topological criticality in arbitrary dimensions, demonstrating that the degeneracy of edge modes can be extracted from the bulk entanglement spectrum. These findings, further supported by numerical simulations of lattice models, provide a universal fingerprint for identifying nontrivial topology in critical free-fermion systems.
Quantum entanglement can be an effective diagnostic tool for probing topological phases protected by global symmetries. Recently, the notion of nontrivial topology in critical systems has been proposed and is attracting growing attention. In this work, as a concrete example, we explore the quantum entanglement properties of fermionic symmetry-enriched quantum critical points by constructing exactly solvable models based on stacked multiple Kitaev chains. We first analytically establish the global phase diagram using entanglement entropy and reveal three topologically distinct gapped phases with different winding numbers, along with three topologically distinct transition lines separating them. Importantly, we unambiguously demonstrate that two transition lines exhibit fundamentally different topological properties despite sharing the same central charge. Specifically, they display nontrivial topological degeneracy in the entanglement spectrum under periodic boundary conditions, thereby generalizing the Li-Haldane bulk-boundary correspondence to a broader class of fermionic symmetry-enriched criticality. Additionally, we identify a novel Lifshitz multicritical point at the intersection of the three transition lines, which also exhibits nontrivial topological degeneracy. This work provides a valuable reference for investigating gapless topological phases of matter from the perspective of quantum entanglement.
We introduce topological invariants for gapless systems and study the associated boundary phenomena.More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality.The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes.We primarily focus on the 1 þ 1d case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors.An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase.An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as 1=L 14 .In addition to such new cases, our formalism unifies various examples previously studied in the literature.Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs.This raises the question of classification, to which we give a partial answer-including a complete characterization of symmetry-enriched 1 þ 1d Ising CFTs.Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 þ 1d CFT without gapped sectors.
Abstract Symmetry-protected topological phases are short-range entangled quantum states characterized by anomalous edge behavior, a manifestation of the bulk-boundary correspondence for topological phases. Moreover, the Li-Haldane conjecture posits that the entanglement spectrum exhibits the same anomaly as the physical edge spectrum, thereby serving as an entanglement-based fingerprint for identifying topological phases. In this work, we extend the entanglement-based diagnostic tools by demonstrating that the edge anomaly is manifested not only in the entanglement spectrum but also in the reduced density matrix itself, a phenomenon we refer to as the mixed-state anomaly. For the two-dimensional SPT phase, we derive a topological contribution in the disorder parameter as a consequence of this mixed-state anomaly. Furthermore, with additional time-reversal symmetry, the anomalous reduced density matrix induces a spontaneous-symmetry-breaking type long-range order in the presence of the disorder operator.