  Amperean superconductivity is an exotic phenomenon stemming from attractive
effective electron-electron interactions (EEEIs) mediated by a transverse gauge
field. Originally introduced in the context of quantum spin liquids and high-Tc
superconductors, Amperean superconductivity has been recently proposed to occur
at temperatures on the order of 1-20 K in two-dimensional, parabolic-band,
electron gases embedded inside deep sub-wavelength optical cavities. In this
work, we first generalize the microscopic theory of cavity-induced Amperean
superconductivity to the case of graphene and then argue that this
superconducting state cannot be achieved in the deep sub-wavelength regime. In
the latter regime, indeed, a cavity induces only EEEIs between density
fluctuations rather than the current-current interactions which are responsible
for Amperean pairing.
