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

TL;DR

This paper analyzes the behavior of the uniform magnetic susceptibility in a Fermi liquid near an antiferromagnetic quantum critical point with dynamic exponent z=2, clarifying theoretical aspects and implications for high-Tc superconductors.

Contribution

It provides a detailed calculation of susceptibility near the critical point, clarifies the role of anomaly graphs, and assesses the applicability of z=2 theories to high-Tc materials.

Findings

Susceptibility behaves as χ(q,0) = χ₀ - D T at the critical point.

The constant D is small and positive in weak coupling.

Behavior in underdoped high-Tc superconductors is hard to explain with z=2 theory.

Abstract

We compute the leading behavior of the uniform magnetic susceptibility, $\chi$, 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, $lim_{q \rightarrow 0} \chi (q,0) = \chi_0 - D T$ with $\chi_0$ 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.