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Studies of the Gr\"uneisen ratio, i.e., the ratio between thermal expansionand specific heat, have become a powerful tool in the context of quantumcriticality, since it was shown theoretically that the Gr\"uneisen ratiodisplays characteristic power-law divergencies upon approaching the transitionpoint of a continuous quantum phase transition. Here we show that theGr\"uneisen ratio also diverges at a symmetry-enhanced first-order quantumphase transition, albeit with mean-field exponents, as the enhanced symmetryimplies the vanishing of a mode gap which is finite away from the transition.We provide explicit results for simple pseudo-spin models, both with andwithout Goldstone modes in the stable phases, and discuss implications.

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