The three central phenomena of cuprate superconductors are linked by a common doping $p^{star}$, where the enigmatic pseudogap phase ends, around which the superconducting phase forms a dome, and at which the resistivity exhibits an anomalous linear dependence on temperature as $T to 0$. However, the fundamental nature of $p^{star}$ remains unclear, in particular whether it marks 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 large enough to suppress superconductivity, over a wide doping range across $p^{star}$. As a function of doping, we find that the electronic term $C_{rm el}$ 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 critical point, as observed in heavy-fermion and iron-based superconductors where their antiferromagnetic phase ends. We conclude that the pseudogap phase of cuprates ends at a quantum critical point, whose associated fluctuations are most likely involved in the $d$-wave pairing and the anomalous scattering.
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