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The three central phenomena of cuprate superconductors are linked by a commondoping $p^{\star}$, where the enigmatic pseudogap phase ends, around which thesuperconducting phase forms a dome, and at which the resistivity exhibits ananomalous linear dependence on temperature as $T \to 0$. However, thefundamental nature of $p^{\star}$ remains unclear, in particular whether itmarks a true quantum phase transition. We have measured the specific heat $C$of the cuprates Eu-LSCO and Nd-LSCO at low temperature in magnetic fields largeenough to suppress superconductivity, over a wide doping range across$p^{\star}$. As a function of doping, we find that the electronic term $C_{\rmel}$ is strongly peaked at $p^{\star}$, where it exhibits a $-T$log$T$dependence as $T \to 0$. These are the classic signatures of a quantum criticalpoint, as observed in heavy-fermion and iron-based superconductors where theirantiferromagnetic phase ends. We conclude that the pseudogap phase of cupratesends at a quantum critical point, whose associated fluctuations are most likelyinvolved in the $d$-wave pairing and the anomalous scattering.

Thermodynamic signatures of quantum criticality in cuprates