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arXiv:2301.07130 · Physical review. B./Physical review. B

Multipartite entanglement in two-dimensional chiral topological liquids

Yuhan Liu, Yuya Kusuki, Jonah Kudler-Flam, Ramanjit Sohal, Shinsei Ryu

Published Feb 6, 2024• 15 citations
arXivHTMLPDF

Abstract

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 $h=\frac{c}{3}ln2$ in the three-vertex state, and generalizes this result to the $p$-vertex state, general rational conformal field theories, and more choices of subsystems. Finally, we support our prediction by numerical evidence, finding precise agreement.

DOI
10.1103/physrevb.109.085108

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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.
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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 $h(A\text{:}B)={S}_{R}(A\text{:}B)\ensuremath{-}I(A\text{:}B)$ 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.
Higher-order topological insulators are materials with a bulk energy gap featuring gapless modes at (d−n)(d-n)(d−n)-dimensional edges, where n>1n > 1n>1. Despite this modified bulk-boundary correspondence, their nontrivial topology can nevertheless be observed through the bipartite entanglement spectrum. Inspired by works suggesting fluctuations in conserved quantities could be used to study topology in one-dimensional systems, which likewise feature zero-dimensional edge modes, we extend this treatment to a two-dimensional second-order topological insulator where corner states play an analogous role. We show that the standard bipartite entanglement entropy and particle number fluctuation lack striking signals of topological phase transitions, both in the static case and in the time evolution after a quench, obscured by an area-law term from the subsystem edges. By introducing a quadripartite construction of the subsystem, we show that it is possible to isolate the topological contributions, producing sharp transition peaks in both quantities, observable by studying subsystems much smaller than the full system. As particle number fluctuations are accessible in existing experiments, these results establish a concrete and scalable path to experimental detection of the entanglement in higher-order topological insulators.
Generally speaking, entanglement entropy (EE) between two subregions of a gapped quantum many-body state is proportional to the area/length of their interface due to the short-range quantum correlation. However, the so-called area law is violated logarithmically in a quantum critical phase. Moreover, the subleading correction exists in long-range entangled topological phases. It is referred to as topological EE which is related to the quantum dimension of the collective excitation in the bulk. Furthermore, if a non-smooth sharp angle is in the presence of the subsystem boundary, a universal angle dependent geometric contribution is expected to appear in the subleading correction. In this work, we simultaneously explore the geometric and edge contributions in the integer quantum Hall (IQH) state and its edge reconstruction in a unified bipartite method. Their scaling is found to be consistent with conformal field theory (CFT) predictions and recent results of particle number fluctuation calculations.
Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.