Role of nematic fluctuations in the thermal melting of pair-density-wave phases in two-dimensional superconductors

We study properties of phase transitions of two-dimensional superconductor liquid crystal phases, and analyze the competition between the recently proposed pair density wave (PDW) and nematic 4e superconductor (4e SC). Nematic fluctuations enhance the 4e SC and suppress the PDW phase. In the absence of lattice effects, the PDW state exists only at T = 0 and the low-temperature phase is a nematic 4e SC with short-ranged PDW order. A geometric description of the 4e SC is presented.