Thermodynamic signature of quantum criticality: universally diverging Gr\"uneisen ratio

At a generic quantum critical point where pressure acts as (or couples to) the zero-temperature control parameter, the Gr\"uneisen ratio $\Gamma$ (the ratio of thermal expansion to specific heat) is {\it divergent}[1]. This property provides a novel probe to quantum criticality from thermodynamics. When scaling applies, $\Gamma \sim 1/T^x$ at the critical pressure $p=p_c$, where the exponent $x$ measures the scaling dimension of the most singular operator coupled to pressure; in the alternative limit $T \to 0$ and $p \neq p_c$, $\Gamma = G_r/(p-p_c)$, where $G_r$ is a universal combination of critical exponents. The predicted divergence has been observed near the quantum critical points of several heavy fermion metals[2]. Analyses based on specific models relevant to these experiments are also presented. [1] L. Zhu, M. Garst, A. Rosch, and Q. Si, Phys. Rev. Lett. {\bf 91}, 066404 (2003). [2] R. K\"uchler {\it et al.}, Phys. Rev. Lett. {\bf 91}, 066405 (2003); {\it ibid.} {\bf 93}, 096402 (2004).