Thermal expansion and Gr\"{u}neisen parameter in quantum Griffiths phases

We consider the behavior of the Gr\"{u}neisen parameter, the ratio between thermal expansion and specific heat, at pressure-tuned infinite-randomness quantum-critical points and in the associated quantum Griffiths phases. We find that the Gr\"{u}neisen parameter diverges as $\ln(T_0/T)$ with vanishing temperature $T$ in the quantum Griffiths phases. At the infinite-randomness critical point itself, the Gr\"{u}neisen parameter behaves as $[\ln(T_0/T)]^{1+1/(\nu\psi)}$ where $\nu$ and $\psi$ are the correlation length and tunneling exponents. Analogous results hold for the magnetocaloric effect at magnetic-field tuned transitions. We contrast clean and dirty systems, we discuss subtle differences between Ising and Heisenberg symmetries, and we relate our findings to recent experiments on CePd$_{1-x}$Rh$_x$.