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The specific heat $C$ of the cuprate superconductors La$_{2-x}$Sr$_x$CuO$_4$and Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$ was measured at lowtemperature (down to $0.5~{\rm K}$), for dopings $p$ close to $p^\star$, thecritical doping for the onset of the pseudogap phase. A magnetic field up to$35~{\rm T}$ was applied to suppress superconductivity, giving direct access tothe normal state at low temperature, and enabling a determination of $C_e$, theelectronic contribution to the normal-state specific heat, at $T \to 0$. InLa$_{2-x}$Sr$_x$CuO$_4$ at $x=p = 0.22$, $0.24$ and $0.25$, $C_e / T =15-16~{\rm mJmol}^{-1}{\rm K}^{-2}$ at $T = 2~{\rm K}$, values that are twiceas large as those measured at higher doping ($p > 0.3$) and lower doping ($p <0.15$). This confirms the presence of a broad peak in the doping dependence of$C_e$ at $p^\star\simeq 0.19$, as previously reported for samples in whichsuperconductivity was destroyed by Zn impurities. Moreover, at those threedopings, we find a logarithmic growth as $T \to 0$, such that $C_e / T \sim{\rm B}\ln(T_0/T)$. The peak vs $p$ and the logarithmic dependence vs $T$ arethe two typical thermodynamic signatures of quantum criticality. In the verydifferent cuprate Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$, we again findthat $C_e / T \sim {\rm B}\ln(T_0/T$) at $p \simeq p^\star$, strong evidencethat this $\ln(1/T)$ dependence - first discovered in the cupratesLa$_{1.8-x}$Eu$_{0.2}$Sr$_x$CuO$_4$ and La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ -is a universal property of the pseudogap critical point. All four materialsdisplay similar values of the $\rm B$ coefficient, indicating that they allbelong to the same universality class.

physical review. b./physical review. b

Normal state specific heat in the cuprates La$_{2-x}$Sr$_x$CuO$_4$ and  Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$ near the critical point of the  pseudogap phase