The specific heat $C$ of the cuprate superconductors ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ and ${\mathrm{Bi}}_{2+y}{\mathrm{Sr}}_{2-x-y}{\mathrm{La}}_x{\mathrm{CuO}}_{6+\delta }$ was measured at low temperatures (down to 0.5 K) for dopings $p$ close to $p^{★}$, the critical doping for the onset of the pseudogap phase. A magnetic field up to 35 T was applied to suppress superconductivity, giving direct access to the normal state at low temperatures, and enabling a determination of $C_{\mathrm{e}}$, the electronic contribution to the normal-state specific heat at $T\to 0$. In ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ at $x=p=0.22$, 0.24 and 0.25, $C_{\mathrm{e}}/T=15\mathrm{to}16\mathrm{mJ}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-2}$ at $T=2\mathrm{K}$, values that are twice as 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 $C_{\mathrm{e}}$ at $p^{★}\simeq 0.19$ as previously reported for samples in which superconductivity was destroyed by Zn impurities. Moreover, at those three dopings, we find a logarithmic growth as $T\to 0$ such that $C_{\mathrm{e}}/T\sim Bln(T_0/T)$. The peak versus $p$ and the logarithmic dependence versus $T$ are the two typical thermodynamic signatures of quantum criticality. In the very different cuprate ${\mathrm{Bi}}_{2+y}{\mathrm{Sr}}_{2-x-y}{\mathrm{La}}_x{\mathrm{CuO}}_{6+\delta }$, we again find that $C_{\mathrm{e}}/T\sim Bln(T_0/T)$ at $p\simeq p^{★}$, strong evidence that this $ln(1/T)$ dependence of the electronic specific heat—first discovered in the cuprates ${\mathrm{La}}_{1.8-x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ and ${\mathrm{La}}_{1.6-x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$—is a universal property of the pseudogap critical point.

• Received 20 January 2021
• Revised 14 May 2021
• Accepted 14 May 2021

DOI:https://doi.org/10.1103/PhysRevB.103.214506