Bipartite Fluctuations of Critical Fermi Surfaces

Fluctuations of conserved quantities within a subsystem are nonlocal 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 ∼𝐿⁢log⁡(𝐿) in Landau Fermi liquids and ℱ ∼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)⁢|𝜎𝑥⁢𝑦|/(2⁢𝜋) when the subsystem geometry contains a sharp corner, where f⁡(𝜃) denotes a function of the corner angle and 𝜎𝑥⁢𝑦 is the Hall conductivity. At the critical point, provided the transition is continuous, the leading scaling ℱ ∼𝐿 is accompanied by a subleading universal corner contribution −log⁡(𝐿)⁢f⁡(𝜃)⁢𝐶𝜌/2 with the same angle dependence f⁡(𝜃), and the universal coefficient 𝐶𝜌 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.