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Topological Multipartite Entanglement in a Fermi Liquid
Mesoscale and Nanoscale Physics · Submitted 13 Apr 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 $\log^D L$ divergence as a function of system size $L$ with a universal coefficient that is proportional to the Euler characteristic $\chi_F$ of the Fermi sea. This provides a generalization, for a Fermi gas, of the well-known result for $D=1$ 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 $\chi_F$. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the $D+1$'th order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to $\chi_F$. Our analytic results are based on the replica method. In addition we perform a numerical study of the charge-weighted mutual information for $D=2$ 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 $\log^3 L$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.81 | 0.95 | 0.79 | asserted 1.00 |
| s1 | 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 $\chi_F$ of the Fermi sea. | result 0.95 | 0.96 | 0.32 | asserted 1.00 |
| s2 | This provides a generalization, for a Fermi gas, of the well-known result for $D=1$ that expresses the $\log L$ divergence of the bipartite entanglement entropy in terms of the central charge $c$ characterizing a conformal field theory. | interpretation 0.72 | 0.44 | 0.11 | asserted 1.00 |
| s3 | For even $D$ we introduce a charge-weighted entanglement entropy that is manifestly odd under a particle-hole transformation. | contribution 0.98 | 0.92 | 0.04 | asserted 0.99 |
| s4 | We show that the corresponding charge-weighted mutual information exhibits a similar $\log^D L$ divergence proportional to $\chi_F$. | result 1.00 | 0.95 | 0.14 | asserted 1.00 |
| s5 | Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the $D+1$'th order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to $\chi_F$. | result 0.55 | 0.93 | 0.16 | asserted 1.00 |
| s6 | Our analytic results are based on the replica method. | method 1.00 | 0.89 | 0.02 | asserted 0.99 |
| s7 | In addition we perform a numerical study of the charge-weighted mutual information for $D=2$ that confirms several aspects of the analytic theory. | contribution 0.52 | 0.97 | 0.03 | asserted 0.94 |
| s8 | Finally, we consider the effect of interactions perturbatively within the replica theory. | contribution 0.64 | 0.92 | 0.02 | asserted 0.50 |
| s9 | We show that for $D=3$ the $\log^3 L$ 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. | result 1.00 | 0.95 | 0.24 | asserted 1.00 |
| s10 | Thus, for $D=3$ the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases. | interpretation 0.97 | 0.93 | 0.48 | asserted 0.99 |
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The paper
Original 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 $\log^D L$ divergence as a function of system size $L$ with a universal coefficient that is proportional to the Euler characteristic $\chi_F$ of the Fermi sea. This provides a generalization, for a Fermi gas, of the well-known result for $D=1$ 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 $\chi_F$. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the $D+1$'th order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to $\chi_F$. Our analytic results are based on the replica method. In addition we perform a numerical study of the charge-weighted mutual information for $D=2$ 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 $\log^3 L$ 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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How the literature uses it
9 arXiv papers cite this in 9 sentences
Papers citing it · 7 sentences without a confident label
Cites it as background 2 papers
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Recently, there has been a burst of theoretical activities aiming at better understanding multipartite entanglement in quantum many-body systems, both in [ref] and out of equilibrium [ref] .
Separability and entanglement of resonating valence-bond states · Introduction · 0.81
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Recently, there is a proposal [ref] for considering charge weighted version of the entanglement entropy.
Information Scrambling and Entanglement Dynamics of Complex Brownian Sachdev-Ye-Kitaev Models · Discussions · 0.85
What it cites · 5 sentences without a confident label
Cites as background 3 papers
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It will be interesting to clarify the connection between our theory of topological entanglement and a recently developed effective field theory of the Fermi surface [ref] that accounts for non-linear effects, and generalizes earlier theories of bosonization of the Fermi surface [ref] which have been applied to study entanglement [ref] .
Nonlinear Bosonization of Fermi Surfaces: The Method of Coadjoint Orbits · Discussion and Conclusion · 0.86
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There has been considerable interest in generalizing this type of analysis to higher dimensions [ref] .
Mutual information and the F-theorem · Introduction · 0.90
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Finally, one of the fascinating things about $c$ in $D=1+1$ is that it can be fractional [ref] .
Applied Conformal Field Theory · Discussion and Conclusion · 0.83
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