Effect of interactions on topology-detecting correlators in Fermi liquids

Fermi surfaces are known to possess nontrivial topology and have been extensively mapped using ARPES and quantum oscillation measurements . However, identifying physical observables that directly reflect Fermi surface topology remains an open problem. Some progress has been made for non-interacting Fermi gases [1,2], where $d+1$ equal-time density--denisty correlations are proportional to the Euler characteristic of the Fermi surface and can thus probe its topology [1]. In this work, we investigate whether correlations of local operators can probe Fermi surface topology in interacting systems. We show that in one-dimensional Luttinger liquids, no correlation of local operators can be constructed that is independent of the interaction parameters, since all operator product expansion (OPE) coefficients and scaling dimensions of primary operators in the Luttinger liquid CFT depend on the interaction strength. As a result, no universal correlation of local operators exists to probe Fermi surface topology in interacting systems. In two-dimensional Fermi liquid theory, we calculate the three-point equal-time density correlation function, which was previously known to probe Fermi surface topology [1] in the non-interacting case, and show that it acquires interaction-dependent corrections, thus losing its universal topology-probing features.