Bipartite Fluctuations of Critical Fermi Surfaces

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-$1\mathrm{/}2$ 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 $-f(\theta )\mathrm{\mid }{\sigma }_{xy}\mathrm{\mid }\mathrm{/}(2\pi )$ when the subsystem geometry contains a sharp corner, where $f(\theta )$ denotes a function of the corner angle, and ${\sigma }_{xy}$ 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 $-\log (L)f(\theta )C_{\rho }\mathrm{/}2$ with the same angle dependence $f(\theta )$, and the universal coefficient $C_{\rho }$ 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.