  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.
