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Physical Review Letters

Singular Three-Point Density Correlations in Two-Dimensional Fermi Liquids

Anonymous, C. L. Kane

Published Jun 29, 2026• 1 citation
Publisher

Abstract

We characterize a singularity in the equal-time three-point density correlations that is generic to two-dimensional interacting Fermi liquids. In momentum space where the three-point correlation is determined by two wave vectors, q_{1} and q_{2}, the singularity takes the form |q_{1}×q_{2}|. We explain how this singularity is sharply defined in a long-wavelength collinear limit. For a noninteracting Fermi gas, the coefficient of this singularity is given by the quantized Euler characteristic of the Fermi sea, and it implies a long-range real space correlation favoring collinear configurations. We show that this singularity persists in interacting Fermi liquids, and express the renormalization of the coefficient of singularity in terms of Landau parameters, for both spinless and spinful Fermi liquids. Our results offer an independent method for measuring Landau parameters. Implications for quantum gas experiments are discussed.

DOI
10.1103/fq7x-srrg

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We characterize a singularity in the equal-time three-point density correlations that is generic to two-dimensional interacting Fermi liquids. In momentum space where the three-point correlation is determined by two wave vectors, q_{1} and q_{2}, the singularity takes the form |q_{1}×q_{2}|. We explain how this singularity is sharply defined in a long-wavelength collinear limit. For a noninteracting Fermi gas, the coefficient of this singularity is given by the quantized Euler characteristic of the Fermi sea, and it implies a long-range real space correlation favoring collinear configurations. We show that this singularity persists in interacting Fermi liquids, and express the renormalization of the coefficient of singularity in terms of Landau parameters, for both spinless and spinful Fermi liquids. Our results offer an independent method for measuring Landau parameters. Implications for quantum gas experiments are discussed.
We characterize a singularity in the equal-time three-point density correlations that is generic to two-dimensional interacting Fermi liquids. In momentum space where the three-point correlation is determined by two wavevectors q1\mathbf{q}_1q1​ and q2\mathbf{q}_2q2​, the singularity takes the form ∣q1×q2∣|\mathbf{q}_1\times\mathbf{q}_2|∣q1​×q2​∣. We explain how this singularity is sharply defined in a long-wavelength collinear limit. For a non-interacting Fermi gas, the coefficient of this singularity is given by the quantized Euler characteristic of the Fermi sea, and it implies a long-range real space correlation favoring collinear configurations. We show that this singularity persists in interacting Fermi liquids, and express the renormalization of the coefficient of singularity in terms of Landau parameters, for both spinless and spinful Fermi liquids. Implications for quantum gas experiments are discussed.
We study the three-point equal-time correlations of the density in a weakly interacting spin 1/2 Fermi gas and present two new results. First, we compute the three-point correlation for the total density ρ=ρ↑+ρ↓ρ= ρ_\uparrow + ρ_\downarrowρ=ρ↑​+ρ↓​ exactly as a function of momentum to first order in a dimensionless interaction parameter I{\cal I}I. This generalizes a previous result that related the three-point function to the Landau Fermi liquid parameters F0sF_0^sF0s​ and F0aF_0^aF0a​ and applied in a certain long-wavelength collinear limit. Second, we compute the leading order O(I3){\cal O}({\cal I}^3)O(I3) interaction correction to the same-spin three-point correlation function in the long-wavelength collinear limit. These results are directly relevant to current experiments on atomic Fermi gases using quantum gas microscopy.
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 DDD-dimensional Fermi sea -- the topological invariant that describes its shape -- is encoded in its (DDD+1)-point density correlations. Here we experimentally demonstrate this connection in a two-dimensional degenerate gas of neutral 6^{6}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.
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 DDD-dimensional Fermi sea is physically encoded in the D+1D+1D+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+1D+1D+1 point density correlation to reveal universal structures of the more general MMM point density correlation functions in a DDD-dimensional Fermi gas. Two experimental methods are proposed for observing these correlations in D=2D=2D=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.
Density three-point correlations are known to probe the topology of the Fermi sea in two-dimensional noninteracting systems. Here, we study how these correlations are modified by interactions using the coadjoint-orbit effective field theory. A key advantage of the coadjoint-orbit formulation is that it provides a systematic way to incorporate generalized Landau interactions in terms of bosonized degrees of freedom, mapping fermionic loop contributions onto simpler tree-level diagrams. We show that, for a general isotropic dispersion ε(p)ε(p)ε(p), even at linear order in the generalized Landau interaction, O(F(2,0))\mathcal{O}(\mathcal{F}^{(2,0)})O(F(2,0)), there exists a contribution proportional to the band curvature ε′′(pF)ε''(p_F)ε′′(pF​) that changes the nonanalytic structure of the free density three-point correlation function. This contribution introduces a distinct nonanalytic structure beyond that found in either the noninteracting case or an interacting Galilean-invariant system, showing that interaction effects can modify the topology-detecting density three-point correlation.