• Open Access

Topological Multipartite Entanglement in a Fermi Liquid

Pok Man Tam, Martin Claassen, and Charles L. Kane
Phys. Rev. X 12, 031022 – Published 2 August 2022

Abstract

We show that the topology of the Fermi sea of a D-dimensional Fermi gas is reflected in the multipartite entanglement characterizing D+1 regions that meet at a point. For odd D we introduce the multipartite mutual information and show that it exhibits a logDL divergence as a function of system size L with a universal coefficient that is proportional to the Euler characteristic χF of the Fermi sea. This provides a generalization, for a Fermi gas, of the well-known result for D=1 that expresses the logL 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 logDL divergence proportional to χF. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the D+1 order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to χF. Our analytic results are based on the replica method. In addition, we perform a numerical study of the charge-weighted mutual information for D=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=3 the log3L divergence of the topological mutual information is not perturbed by weak short-ranged interactions, though for D=2 the charge-weighted mutual information is perturbed. Thus, for D=3 the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases.

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  • Received 19 April 2022
  • Accepted 12 July 2022

DOI:https://doi.org/10.1103/PhysRevX.12.031022

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Pok Man Tam, Martin Claassen, and Charles L. Kane*

  • Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

  • *Corresponding author. kane@physics.upenn.edu

Popular Summary

Two powerful principles for characterizing the structure of complex quantum states are topology and entanglement. Topology reveals the universal features of quantum states that are insensitive to smooth deformations, while entanglement describes intrinsic nonlocal correlations among particles. Interrelations between topology and entanglement have been found for some classes of quantum states, but a general theory has remained elusive. In this work, we establish a new connection between these concepts for the many-electron quantum states of metals.

The quantum ground state of a metal is characterized by an abstract surface in momentum space called the Fermi surface. These surfaces can be distinguished topologically in much the same way that a sphere can be distinguished from a donut by counting the number of holes. Every Fermi surface has an integer topological invariant called the Euler characteristic, which is related to its number of holes. This work establishes that the Euler characteristic of the Fermi surface is related to a measure of entanglement called the mutual information, which characterizes the correlations between four regions in real space that meet at a single point. The mutual information exhibits a universal logarithmic divergence as a function of system size with a coefficient proportional to the Euler characteristic of the Fermi surface.

This result provides a starting point for characterizing the connection between topology and entanglement in strongly interacting phases and at critical points, and it introduces concepts that may be useful for developing experimental probes of Fermi surface topology.

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Vol. 12, Iss. 3 — July - September 2022

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