  When Fermi surfaces (FSs) are subject to long-range interactions that are
marginal in the renormalization-group sense, Landau Fermi liquids are
destroyed, but only barely. With the interaction further screened by
particle-hole excitations through one-loop quantum corrections, it has been
believed that these marginal Fermi liquids (MFLs) are described by weakly
coupled field theories at low energies. In this Letter, we point out a
possibility in which higher-loop processes qualitatively change the picture
through UV-IR mixing, in which the size of the FS enters as a relevant scale.
The UV-IR mixing effect enhances the coupling at low energies, such that the
basin of attraction for the weakly coupled fixed point of a $(2+1)$-dimensional
MFL shrinks to a measure-zero set in the low-energy limit. This UV-IR mixing is
caused by gapless virtual Cooper pairs that spread over the entire FS through
marginal long-range interactions. Our finding signals a possible breakdown of
the patch description for the MFL and questions the validity of using the MFL
as the base theory in a controlled scheme for non-Fermi liquids that arise from
relevant long-range interactions.
