TY - JOUR

T1 - Multipartitioning topological phases by vertex states and quantum entanglement

AU - Liu, Yuhan

AU - Sohal, Ramanjit

AU - Kudler-Flam, Jonah

AU - Ryu, Shinsei

N1 - Funding Information:
We would like to acknowledge R. Mong, K. Siva, T. Soejima, M. Zaletel, and Y. Zou for insightful discussions, and for sharing their manuscript prior to arXiv submission. S.R. is supported by the National Science Foundation under Award No. DMR-2001181, and by a Simons Investigator Grant from the Simons Foundation (Award No. 566116). R.S. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (funding Ref. No. 6799-516762-2018) and of the U.S. National Science Foundation under Grant No. DMR-1725401 at the University of Illinois.
Publisher Copyright:
© 2022 American Physical Society.

PY - 2022/3/15

Y1 - 2022/3/15

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevB.105.115107

DO - 10.1103/PhysRevB.105.115107

M3 - Article

AN - SCOPUS:85126428477

SN - 2469-9950

VL - 105

JO - Physical Review B

JF - Physical Review B

IS - 11

M1 - 115107

ER -