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

At a generic quantum critical point, the thermal expansion $\ensuremath{\alpha}$ is more singular than the specific heat ${c}_{p}$. Consequently, the ``Gr\"uneisen ratio,'' $\ensuremath{\Gamma}=\ensuremath{\alpha}/{c}_{p}$, diverges. When scaling applies, $\ensuremath{\Gamma}\ensuremath{\sim}{T}^{\ensuremath{-}1/(\ensuremath{\nu}z)}$ 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 $T\ensuremath{\rightarrow}0$ and $p\ensuremath{\ne}{p}_{c}$, $\ensuremath{\Gamma}\ensuremath{\sim}1/(p\ensuremath{-}{p}_{c})$ 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 $(1/T)\ensuremath{\partial}T/\ensuremath{\partial}H{|}_{S}$. Finally, we determine the corrections to scaling in a class of metallic quantum critical points.

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