On the Critical Behavior of the Uniform Susceptibility of a Fermi Liquid Near an Antiferromagnetic Transition with Dynamic Exponent $ z = 2$

Abstract

We compute the leading behavior of the uniform magnetic susceptibility, $χ$, of a Fermi liquid near an antiferromagnetic transition with dynamic exponent $z=2$. Our calculation clarifies the role of triangular ``anomaly'' graphs in the theory and justifies the effective action used in previous work \cite{Hertz}. We find that at the $z=2$ critical point of a two dimensional material, $lim_{q \rightarrow 0} χ(q,0) = χ_0 - D T$ with $χ_0$ and $D$ nonuniversal constants. For reasonable band structures we find that in a weak coupling approximation $D$ is small and positive. Our result suggests that the behavior observed in the quantum critical regime of underdoped high-$T_c$ superconductors are difficult to explain in a $z=2$ theory.