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arXiv:2110.11980 · Physical review. B./Physical review. B

Multipartitioning topological phases by vertex states and quantum entanglement

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

Published Mar 4, 2022• 30 citations
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Abstract

We discuss multipartitions of the gapped ground states of $(2+1)$-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 $(1+1)$-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 $(c/3)ln2$ where $c$ is the central charge of the topological liquid that measures ungappable edge degrees of freedom. As specific examples, we consider topological chiral $p$-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.

DOI
10.1103/physrevb.105.115107

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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.
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.
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.
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.
Multi-wavefunction overlaps -- generalizations of the quantum mechanical inner product for more than two quantum many-body states -- are valuable tools for studying many-body physics. In this paper, we investigate the multi-wavefunction overlap of (2+1)-dimensional gapped ground states, focusing particularly on symmetry-protected topological (SPT) states. We demonstrate how these overlaps can be calculated using the bulk-boundary correspondence and (1+1)-dimensional edge theories, specifically conformal field theory. When applied to SPT phases, we show that the topological invariants, which can be thought of as discrete higher Berry phases, can be extracted from the multi-wavefunction overlap of four ground states with appropriate symmetry actions. Additionally, we find that the multi-wavefunction overlap can be expressed in terms of the realignment of reduced density matrices. Furthermore, we illustrate that the same technique can be used to evaluate the multi-entropy -- a quantum information theoretical quantity associated with multi-partition of many-body quantum states -- for (2+1)-dimensional gapped ground states. Combined with numerics, we show that the difference between multi-entropy for tripartition and second Rényi entropies is bounded from below by (ctot/4)ln⁡2(c_{\it tot}/4)\ln 2(ctot​/4)ln2 where ctotc_{\it tot}ctot​ is the central charge of ungappable degrees of freedom. To calculate multi-entropy numerically for free fermion systems (such as Chern insulators), we develop the correlator method for multi-entropy.
We show that the topology of the Fermi sea of a D-dimensional Fermi gas is reflected in the multipartite entanglement characterizing D þ 1 regions that meet at a point.For odd D we introduce the multipartite mutual information and show that it exhibits a log D L divergence as a function of system size L with a universal coefficient that is proportional to the Euler characteristic χ F of the Fermi sea.This provides a generalization, for a Fermi gas, of the well-known result for D ¼ 1 that expresses the log L divergence of the bipartite entanglement entropy in terms of the central charge c characterizing a conformal field theory.For even D we introduce a charge-weighted entanglement entropy that is manifestly odd under a particle-hole transformation.We show that the corresponding charge-weighted mutual information exhibits a similar log D L divergence proportional to χ F .Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the D þ 1 order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to χ F .Our analytic results are based on the replica method.In addition, we perform a numerical study of the charge-weighted mutual information for D ¼ 2 that confirms several aspects of the analytic theory.Finally, we consider the effect of interactions perturbatively within the replica theory.We show that for D ¼ 3 the log 3 L divergence of the topological mutual information is not perturbed by weak short-ranged interactions, though for D ¼ 2 the charge-weighted mutual information is perturbed.Thus, for D ¼ 3 the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases.
Boundaries constitute a rich playground for quantum many-body systems because they can lead to novel degrees of freedom such as protected boundary states in topological phases. Here, we study the ground state of integer quantum Hall systems in the presence of boundaries through the reduced density matrix of a spatial region. We work in the lowest Landau level and choose our region to intersect the boundary at arbitrary angles. The entanglement entropy (EE) contains a logarithmic contribution coming from the chiral edge modes, and matches the corresponding conformal field theory prediction. We uncover an additional contribution due to the boundary corners. We characterize the angle dependence of this boundary corner term, and compare it to the bulk corner EE. We further analyze the spatial structure of entanglement via the eigenstates associated with the reduced density matrix. The influence of the physical boundary and the region's geometry on the reduced density matrix is thus clarified. Finally, we test a bulk-boundary correspondence for the EE originally obtained for quantum critical systems such as conformal field theories in two spatial dimensions, and discuss the implications of our findings for other topological phases.