Divergence of the Gr\"uneisen ratio at symmetry-enhanced first-order quantum phase transitions

Studies of the Gr"uneisen ratio, i.e., the ratio between thermal expansion and specific heat, have become a powerful tool in the context of quantum criticality, since it was shown theoretically that the Gr"uneisen ratio displays characteristic power-law divergencies upon approaching the transition point of a continuous quantum phase transition. Here we show that the Gr"uneisen ratio also diverges at a symmetry-enhanced first-order quantum phase transition, albeit with mean-field exponents, as the enhanced symmetry implies the vanishing of a mode gap which is finite away from the transition. We provide explicit results for simple pseudo-spin models, both with and without Goldstone modes in the stable phases, and discuss implications.