Universally Diverging Grüneisen Parameter and the Magnetocaloric Effect Close to Quantum Critical Points

At a generic quantum critical point, the thermal expansion $\alpha$ is more singular than the specific heat $c_p$. Consequently, the “Grüneisen ratio,” $\Gamma =\alpha /c_p$, diverges. When scaling applies, $\Gamma \sim T^{-1/(\nu z)}$ at the critical pressure $p=p_c$, providing a means to measure the scaling dimension of the most relevant operator that pressure couples to; in the alternative limit $T\to 0$ and $p\ne p_c$, $\Gamma \sim 1/(p-p_c)$ with a prefactor that is, up to the molar volume, a simple universal combination of critical exponents. For a magnetic-field driven transition, similar relations hold for the magnetocaloric effect $(1/T)\partial T/\partial H{|}_S$. Finally, we determine the corrections to scaling in a class of metallic quantum critical points.