Distinguishing XY from Ising electron nematics

At low temperatures in ultraclean GaAs-AlGaAs heterojunctions, high Landau levels near half-integral filling break rotational symmetry, leading to increasingly anisotropic transport properties as temperature is lowered below similar to 150 mK. While the onset of transport anisotropy is well described by an XY model of an electron nematic in the presence of a weak uniform symmetry-breaking term, the low-temperature behavior deviates significantly from this model. We find that inclusion of interactions between the electron nematic and the underlying crystalline lattice in the form of a fourfold symmetry-breaking term is sufficient to describe the entire temperature dependence of the transport anisotropy at nu = 9/2. This implies that this electron nematic is in the Ising universality class. We propose new experimental tests that can distinguish whether any two-dimensional electron nematic is in the XY or Ising universality class.