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

At a generic quantum critical point, the thermal expansion α is more singular than the specific heat cₚ. Consequently, the ``Gr\"uneisen ratio,'' Γ=α/cₚ, diverges. When scaling applies, Γ~T^-1/(νz) at the critical pressure p=pc, providing a means to measure the scaling dimension of the most relevant operator that pressure couples to; in the alternative limit T→0 and p≠pc, Γ~1/(p-pc) 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)∂T/∂H|S. Finally, we determine the corrections to scaling in a class of metallic quantum critical points.