Effect of the Number of Hole Pockets on the Charge-Orbital Nematic Order in FeSe_1-xS_x System

We investigate if there is a novel way to distinguish between two different types of nematic orders that exist in systems with multiple degrees of freedom, e.g Fe based superconductors: the charge-orbital Pomeranchuk order and composite spin-nematic order. Both of them break the same Z2 symmetry. We calculate how the onset temperature for each of these two orders behaves near a quantum critical point for nematicity. Common wisdom says that, near a quantum critical point, they vary as power law. Surprisingly, we find that for Pomeranchuk order the result is more involved. Namely, in the presence of two hole pockets, there is a perfect cancellation of the power law contributions, and the transition temperature Tp scales as log δ, where δ is the deviation from the quantum critical point. Once the inner hole pocket sinks below the Fermi level due to e.g., spin-orbit coupling, power-law dependence Tp ~ δ^1/2 is restored. At the border line, when the inner hole pocket just vanishes, we find a linear dependence Tp ~ δ. For composite spin order the Tp ~ δ^1/2 dependence holds independent on the geometry of hole pockets. We propose to use these results as a tool to distinguish between Pomeranchuk and composite spin-nematic order.