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arXiv:2110.11965 · Physical review. B./Physical review. B

Universal tripartite entanglement signature of ungappable edge states

Karthik Siva, Yijian Zou, Tomohiro Soejima, Roger S. K. Mong, Michael P. Zaletel

Published Jul 11, 2022• 25 citations
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Abstract

Gapped two-dimensional topological phases can feature ungappable edge states which are robust even in the absence of protecting symmetries. In this Letter we argue that a multipartite entanglement measure recently proposed in the context of holography, the Markov gap, provides a universal diagnostic of ungappable edge states. Defined as a difference of the reflected entropy and mutual information $h(A\text{:}B)={S}_{R}(A\text{:}B)\ensuremath{-}I(A\text{:}B)$ between two parties, we argue that for $A,B$ being adjacent subregions in the bulk $h=\frac{{c}_{+}}{3}ln\phantom{\rule{0.16em}{0ex}}2$, where ${c}_{+}$ is the minimal total central charge of the boundary theory. As evidence, we prove that $h=0$ for string-net models and numerically verify that $h=\frac{|C|}{3}ln\phantom{\rule{0.16em}{0ex}}2$ for a Chern-$C$ insulator. Our Letter establishes a unique bulk entanglement criteria for the presence of a conformal field theory on the boundary.

DOI
10.1103/physrevb.106.l041107

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Gapped two-dimensional topological phases can feature ungappable edge states which are robust even in the absence of protecting symmetries. In this Letter we argue that a multipartite entanglement measure recently proposed in the context of holography, the Markov gap, provides a universal diagnostic of ungappable edge states. Defined as a difference of the reflected entropy and mutual information h(A:B)=SR(A:B)\ensuremath−I(A:B)h(A\text{:}B)={S}_{R}(A\text{:}B)\ensuremath{-}I(A\text{:}B)h(A:B)=SR​(A:B)\ensuremath−I(A:B) between two parties, we argue that for A,BA,BA,B being adjacent subregions in the bulk h=c+3ln2h=\frac{{c}_{+}}{3}ln\phantom{\rule{0.16em}{0ex}}2h=3c+​​ln2, where c+{c}_{+}c+​ is the minimal total central charge of the boundary theory. As evidence, we prove that h=0h=0h=0 for string-net models and numerically verify that h=∣C∣3ln2h=\frac{|C|}{3}ln\phantom{\rule{0.16em}{0ex}}2h=3∣C∣​ln2 for a Chern-CCC insulator. Our Letter establishes a unique bulk entanglement criteria for the presence of a conformal field theory on the boundary.
We discuss multipartitions of the gapped ground states of (2+1)(2+1)(2+1)-dimensional topological liquids into three (or more) spatial regions that are adjacent to each other and meet at points. By considering the reduced density matrix obtained by tracing over a subset of the regions, we compute various correlation measures, such as entanglement negativity, reflected entropy, and associated spectra. We utilize the bulk-boundary correspondence to show that such multipartitions can be achieved by using what we call vertex states in (1+1)(1+1)(1+1)-dimensional conformal field theory: these are a type of state used to define an interaction vertex in string field theory and can be thought of as a proper generalization of conformal boundary states. This approach allows an explicit construction of the reduced density matrix near the entangling boundaries. We find the fingerprints of topological liquid in these quantities, such as (universal pieces in) the scaling of the entanglement negativity, and a nontrivial distribution of the spectrum of the partially transposed density matrix. For reflected entropy, we test the recent claim that states the difference between reflected entropy and mutual information is given, once short-range correlations are properly removed, by (c/3)ln2(c/3)ln2(c/3)ln2 where ccc is the central charge of the topological liquid that measures ungappable edge degrees of freedom. As specific examples, we consider topological chiral ppp-wave superconductors and Chern insulators. We also study a specific lattice-fermion model realizing Chern insulator phases and calculate the correlation measures numerically, both in its gapped phases and at critical points separating them.
The multipartite entanglement structure for the ground states of two-dimensional (2D) topological phases is an interesting albeit not well-understood question. Utilizing the bulk-boundary correspondence, the calculation of tripartite entanglement in 2D topological phases can be reduced to that of the vertex state, defined by the boundary conditions at the interfaces between spatial regions. In this paper, we use the conformal interface technique to calculate entanglement measures in the vertex state, which include area-law terms, corner contributions, and topological pieces, and a possible additional order-one contribution. This explains our previous observation of the Markov gap h=c3ln2h=\frac{c}{3}ln2h=3c​ln2 in the three-vertex state, and generalizes this result to the ppp-vertex state, general rational conformal field theories, and more choices of subsystems. Finally, we support our prediction by numerical evidence, finding precise agreement.
Motivated by conjectures in holography relating the entanglement of purification and reflected entropy to the entanglement wedge cross section, we introduce two related non-negative measures of tripartite entanglement g and h. We prove structure theorems which show that states with nonzero g or h have nontrivial tripartite entanglement. We then establish that in one dimension these tripartite entanglement measures are universal quantities that depend only on the emergent low-energy theory. For a gapped system, we argue that either g≠0 and h=0 or g=h=0, depending on whether the ground state has long-range order. For a critical system, we develop a numerical algorithm for computing g and h from a lattice model. We compute g and h for various CFTs and show that h depends only on the central charge whereas g depends on the whole operator content.
A bstract The reflected entropy S R ( A : B ) of a density matrix ρ AB is a bipartite correlation measure lower-bounded by the quantum mutual information I ( A : B ). In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order- N 2 gap between S R and I . We provide an information-theoretic interpretation of this gap by observing that S R − I is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity S R − I the Markov gap . We then prove that for time-symmetric states in pure AdS 3 gravity, the Markov gap is universally lower bounded by log(2) ℓ AdS / 2 G N times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling S R − I using fixed area states. This analysis involves deriving a formula for the quantum fidelity — in fact, for all the sandwiched Rényi relative entropies — between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.
The hallmark of two-dimensional chiral topological phases is the existence of anomalous gapless modes at the spatial boundary. Yet, the manifestation of this edge anomaly within the bulk ground-state wavefunction itself remains only partially understood. In this work, we introduce a family of multipartite entanglement measures that probe chirality directly from the bulk wavefunction. Our construction involves applying different permutations between replicas of the ground state wavefunction in neighboring spatial regions, creating "permutation defects" at the boundaries between these regions. We provide general arguments for the robustness of these measures and develop a field-theoretical framework to compute them systematically. While the standard topological field theory prescription misses the chiral contribution, our method correctly identifies it as the chiral conformal field theory partition function on high-genus Riemann surfaces. This feature is a consequence of the bulk-edge correspondence, which dictates that any regularization of the theory at the permutation defects must introduce gapless boundary modes. We numerically verify our results with both free-fermion and strongly-interacting chiral topological states and find excellent agreement. Our results enable the extraction of the chiral central charge and the Hall conductance using a finite number of wavefunction replicas, making these quantities accessible to Monte-Carlo numerical techniques and noisy intermediate-scale quantum devices.
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.