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.
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.
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.
Motivated by recent development of the concept of the disorder operator and its relation with entanglement entropy in bosonic systems, here we show the disorder operator successfully probes many aspects of quantum entanglement in fermionic many-body systems. From both analytical and numerical computations in free and interacting fermion systems in 1D and 2D, we find the disorder operator and the entanglement entropy exhibit similar universal scaling behavior, as a function of the boundary length of the subsystem, but with subtle yet important differences. In 1D they both follow the \log{L} log L scaling behavior with the coefficient determined by the Luttinger parameter for disorder operator, and the conformal central charge for entanglement entropy. In 2D they both show the universal L\log L L log L scaling behavior in free and interacting Fermi liquid states, with the coefficients depending on the geometry of the Fermi surfaces. However at a 2D quantum critical point with non-Fermi-liquid state, extra symmetry information is needed in the design of the disorder operator, so as to reveal the critical fluctuations as does the entanglement entropy. Our results demonstrate the fermion disorder operator can be used to probe quantum many-body entanglement related to global symmetry, and provide new tools to explore the still largely unknown territory of highly entangled fermion quantum matter in 2 or higher dimensions.
A Fermi gas of noninteracting electrons, or ultracold fermionic atoms, has a quantum ground state defined by a region of occupancy in momentum space known as the Fermi sea. The Euler characteristic ${\ensuremath{\chi}}_{F}$ of the Fermi sea serves to topologically classify these gapless fermionic states. The topology of a D-dimensional Fermi sea is physically encoded in the D+1 point equal time density correlation function. In this work, we first present a simple proof of this fact by showing that the evaluation of the correlation function can be formulated in terms of a triangulation of the Fermi sea with a collection of points, links, and triangles and their higher dimensional analogs. We then make use of the topological D+1 point density correlation to reveal universal structures of the more general M point density correlation functions in a D-dimensional Fermi gas. Two experimental methods are proposed for observing these correlations in D=2. In cold atomic gases imaged by quantum gas microscopy, our analysis supports the feasibility of measuring the third-order density correlation, from which ${\ensuremath{\chi}}_{F}$ can be reliably extracted in systems with as few as around 100 atoms. For solid-state electron gases, we propose measuring correlations in the speckle pattern of intensity fluctuations in nonlinear x-ray scattering experiments.
Pauli's exclusion principle forces fermions to occupy distinct quantum states, creating a filled region of momentum space at low temperature, the Fermi sea, whose topology governs the system's response to perturbations and the nature of its correlation functions. Recent theory predicts that for non-interacting fermions, the Euler characteristic of a D-dimensional Fermi sea -- the topological invariant that describes its shape -- is encoded in its (D+1)-point density correlations. Here we experimentally demonstrate this connection in a two-dimensional degenerate gas of neutral 6Li atoms using single-atom-resolved imaging. By measuring three- and four-point connected density correlations in real space, we directly extract topological invariants of the underlying Fermi sea, including the Euler characteristic. Our results are in remarkable agreement with ideal-gas predictions, despite the presence of sizeable interactions, and establish a new pathway for probing many-body topology through correlation measurements.
Abstract We study different aspects of quantum field theory at finite density using methods from quantum information theory. For simplicity we focus on massive Dirac fermions with nonzero chemical potential, and work in 1 + 1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. Unlike what happens in Lorentz-invariant theories, this c-function exhibits a strong violation of monotonicity; it also encodes the creation of long-range entanglement from the Fermi surface. Motivated by previous works on lattice models, we next calculate numerically the Renyi entropies and find Friedel-type oscillations; these are understood in terms of a defect operator product expansion. Furthermore, we consider the mutual information as a measure of correlation functions between different regions. Using a long-distance expansion previously developed by Cardy, we argue that the mutual information detects Fermi surface correlations already at leading order in the expansion. We also analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge and/or mass. In particular, we show that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy. Finally, we discuss possible extensions to interacting theories, and argue for the relevance of some of these measures for probing non-Fermi liquids.
Bipartite fluctuations can provide interesting information about entanglement properties and correlations in many-body quantum systems. We address such fluctuations in relation with the topology of Dirac and Weyl quantum systems, in situations where the relevant particle number is not conserved, leading to additional volume laws scaling with the quantum Fisher information. In the case of the p+ip superconductor, we build a relation between charge fluctuations and the associated winding numbers of Dirac cones in the low-energy sector. Topological aspects of the Hamiltonian in the vicinity of these points induce long-range entanglement in real space. We specifically extract analytical expressions for the corner contributions to bipartite fluctuations. We highlight their differences and similarities with the corner functions of the entanglement entropy which characterize the underlying conformal field theory. We provide a detailed analysis of such fluctuation properties, including the role of gap anisotropy, and discuss higher-dimensional Weyl analogs.