Upper bound on the window of density occupied by microemulsion phases in two-dimensional electron systems

In two-dimensional electronic systems, direct first-order phase transitions are prohibited as a consequence of the long-range Coulomb interaction, which implies a stiff energetic penalty for macroscopic phase separation. A prominent proposal is that any direct first-order transition is instead replaced by a sequence of ``microemulsion" phases, in which the two phases are mixed in patterns of mesoscopic domains. In this note, we comment on the range $\mathrm{\Delta }n$ of average electron density that such microemulsion phases may occupy. We point out that, even without knowing the value of a phenomenological parameter associated with surface tension between the two phases, one can place a fairly strong upper bound on the value of $\mathrm{\Delta }n$. We make numerical estimates for $\mathrm{\Delta }n$ in the case of the Fermi liquid to Wigner crystal transition and find $\mathrm{\Delta }n$ to be on the order of ${10}^7$,cm${}^{-2}$. This value is much smaller than the width of the phase transition observed in experiments, suggesting that disorder is a more likely explanation for the apparent broadening of the transition.