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arXiv:2608.28588

Interaction corrections to topological density three-point functions in two-dimensional Fermi liquids: a coadjoint orbit perspective

Akshay Pal, Andrew Lucas, Umang Mehta

Published Aug 28, 2026• 0 citations
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Abstract

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 $ε(p)$, even at linear order in the generalized Landau interaction, $\mathcal{O}(\mathcal{F}^{(2,0)})$, there exists a contribution proportional to the band curvature $ε''(p_F)$ 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.

DOI
10.48550/arxiv.2608.28588

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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.
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.
We develop a method for bosonizing the Fermi surface based on the formalism of the coadjoint orbits. This allows one to parametrize the Fermi surface by a bosonic field that depends on the spacetime coordinates and on the position on the Fermi surface. The Wess-Zumino-Witten term in the effective action, governing the adiabatic phase acquired when the Fermi surface changes its shape, is completely fixed. As an effective field theory the action also involves a Hamiltonian, which contains, beside the kinetic energy and the Landau interaction, terms with arbitrary number of derivatives and fields. We show that the resulting local effective field theory captures both linear and nonlinear effects in Landau's Fermi liquid theory. The approach can be extended to incorporate spin degrees of freedom and the charge-2 fields corresponding to the BCS order parameter.
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 present a bosonized effective field theory for a 2d Fermi surface in a weak magnetic field using the coadjoint orbit approach, which was recently developed as a nonlinear bosonization method in phase space for Fermi liquids and non-Fermi liquids. We show that by parametrizing the phase space with the guiding center and the mechanical momentum, and by using techniques in noncommutative field theory, the physics of Landau levels and Landau level degeneracy (NΦN_ΦNΦ​) naturally arises. For a parabolic dispersion, the resulting theory describes NΦN_ΦNΦ​ flavors of free chiral bosons propagating in \emph{momentum space}. In addition, the action contains a linear term in the bosonic field, which upon mode expansion becomes a topological θθθ-term. By properly quantizing this theory, we reproduce the well-known thermal and magnetic responses of a Fermi surface, including linear-in-TTT specific heat, Landau diamagnetism, and the de Haas-van Alphen effect. {In particular, the de Haas-van Alphen effect is shown to be a direct consequence of the topological θθθ-term.} Our theory paves the way toward understanding correlated gapless fermionic systems in a magnetic field using the powerful approach of bosonization.
The discovery of many strongly correlated metallic phases has inspired different routes to generalize or go beyond the celebrated Landau Fermi liquid theory. To this end, from universal consideration of symmetries and anomalies, Else, Thorngren, and Senthil (ETS) have introduced a class of theories called ersatz Fermi liquids which possess a Fermi surface and satisfy a generalized Luttinger's theorem. In this work, we view all such fermion liquids obeying the Luttinger theorem as incompressible quantum Hall liquids in higher-dimensional phase space and use it as the starting point to derive their effective low-energy field theory. The noncommutativity of phase space motivates us to use the Seiberg-Witten map to derive the field theory in an ordinary (commutative) space and naturally leads to terms that correspond to the correct topological Chern-Simons action postulated by ETS in one, two, and three dimensions. Additionally, our approach also reproduces all the nontopological terms that characterize important contributions to the response, including the semiclassical equations of motion. Finally, our derivations of Chern-Simons terms from the Seiberg-Witten map also verify a longstanding conjecture in noncommutative field theory.
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.