Singular order parameter interaction at the nematic quantum critical point in two-dimensional electron systems

We analyze the infrared behavior of effective $N$-point interactions between order parameter fluctuations for nematic and other quantum critical electron systems with a scalar order parameter in two dimensions. The interactions exhibit a singular momentum and energy dependence and thus can not be represented by local vertices. They diverge for all $N\ge 4$ in a collinear low-energy limit, where energy variables scale to zero faster than momenta, and momenta become increasingly collinear. The degree of divergence is not reduced by any cancellations and renders all $N$-point interactions marginal. A truncation of the order parameter action at quartic or any other finite order is therefore not justified. The same conclusion can be drawn for the effective action describing fermions coupled to a U$\left(1\right)$ gauge field in two dimensions.

Figure 1

Graphical representation of ${\Pi }_{d,N}$ with momentum variables as in Eq. (3).