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arXiv:2607.05747

Three-point density correlations in a weakly interacting 2D Fermi liquid

C. L. Kane

Published Jul 7, 2026• 0 citations
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Abstract

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 $ρ= ρ_\uparrow + ρ_\downarrow$ exactly as a function of momentum to first order in a dimensionless interaction parameter ${\cal I}$. This generalizes a previous result that related the three-point function to the Landau Fermi liquid parameters $F_0^s$ and $F_0^a$ and applied in a certain long-wavelength collinear limit. Second, we compute the leading order ${\cal O}({\cal I}^3)$ 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.

DOI
10.48550/arxiv.2607.05747

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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.
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 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.
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.
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We derive exact general relations between various observables for NNN spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range re{r}_{e}re​ in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading 1/k61/{k}^{6}1/k6 tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-1/a1/a1/a and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.
We study a weak interaction quench in a three-dimensional Fermi gas. We first show that, under some general assumptions on time-dependent perturbation theory, the perturbative expansion of the long-wavelength structure factor S(q)S(\mathbf{q})S(q) is not compatible with the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q)S(\mathbf{q})S(q) does develop an analytical component $\ensuremath{\sim}\mathrm{const}+O({q}^{2})$ at $\mathbf{q}\ensuremath{\rightarrow}\mathbf{0}$, as implied by thermalization, but, in contrast, it maintains a nonanalytic part $\ensuremath{\sim}|\mathbf{q}|$ characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity corresponds to persisting power-law decaying density-density correlations, whereas thermalization would predict only an exponential decay. We next consider the case of a dilute gas, where one can obtain nonperturbative results in the interaction strength but at lowest order in the density. We find that in the steady state the momentum distribution jump at the Fermi surface remains finite, though smaller than in equilibrium, up to second order in kFf0{k}_{F}{f}_{0}kF​f0​, where f0{f}_{0}f0​ is the scattering length of two particles in the vacuum. Both results question the emergence of a finite length scale in the quench dynamics as expected by thermalization.