Bipartite Fluctuations of Critical Fermi Surfaces

Abstract

Fluctuations of conserved quantities within a subsystem are non-local observables that provide unique insights into quantum many-body systems. In this paper, we study bipartite charge (and spin) fluctuations across interaction-driven ``metal-insulator transitions'' out of Landau Fermi liquids. The ``charge insulators'' include a class of non-Fermi-liquid states of fractionalized degrees of freedom, such as compressible composite Fermi liquids (for spinless electrons) and incompressible spin-liquid Mott insulators (for spin-$1/2$ electrons). We find that charge fluctuations $F$ exhibit distinct leading-order scalings across the transition: $F \sim L\log(L)$ in Landau Fermi liquids and $F \sim L$ in charge insulators, where $L$ is the linear size of the subsystem. In composite Fermi liquids, under certain conditions, we also identify a universal constant term $-f(θ)|σ_{xy}|/(2π)$ when the subsystem geometry contains a sharp corner, where $f(θ)$ denotes a function of the corner angle, and $σ_{xy}$ is the Hall conductivity. At the critical point, provided the transition is continuous, the leading scaling $F\sim L$ is accompanied by a subleading universal corner contribution $-\log(L)f(θ)C_ρ/2$ with the same angle dependence $f(θ)$, and the universal coefficient $C_ρ$ is directly related to the predicted universal jumps in longitudinal and Hall resistivities. These results establish fluctuation-transport relations, paving the way for numerical and experimental studies of unconventional quantum criticalities in metals.