How prominent are microemulsion phases in 2D electron systems?

In two-dimensional electronic systems, direct first-order phase transitions are prohibited by the long-range Coulomb interaction, which implies a huge energy penalty for macroscopic phase separation. A prominent proposal is that direct first-order transitions are replaced by a series of "microemulsion" phases characterized by patterns of mesoscopic domains where the two phases are intermixed. In this work, we examine the range (Δn) of average electron density that these microemulsion phases may occupy. We demonstrate that, even without knowing the value of a phenomenological surface tension parameter, one can establish a robust upper bound for Δn. This result has wide-ranging implications for interpreting experiments. For the Wigner crystal - Fermi liquid transition, we derive a remarkably narrow bound, which helps to explain why microemulsion phases are not observed in quantum Monte Carlo studies.