High density of states in the pseudogap phase of the cuprate superconductor HgBa${}_2$CuO${}_{4+\delta }$

The specific heat $C$ of the single-layer cuprate superconductor HgBa${}_2$CuO${}_{4+\delta }$ was measured in an underdoped crystal with $T_{\rm c} = 72$Kattemperaturesdownto$2$Kinmagneticfieldsupto$35$ T, a field large enough to suppress superconductivity at that doping ($p \simeq 0.09$)\mathrm{.}Inthenormalstateat$H = 35$T,aresiduallineartermofmagnitude$\gamma = 12 \pm 2$mJ\mathrm{/}K$^2$molisobservedin$C/T$as$T \to 0$, a direct measure of the electronic density of states. This high value of $\gamma$ has two major implications. First, it is significantly larger than the value measured in overdoped cuprates outside the pseudogap phase ($p>p^{\star }$), such as La${}_{2-x}$Sr${}_x$CuO${}_4$ and Tl${}_2$Ba${}_2$CuO${}_{6+\delta }$ at $p \simeq 0.3$,where$\gamma \simeq 7$mJ\mathrm{/}K$^2$mol. Given that the pseudogap causes a loss of density of states, and assuming that HgBa${}_2$CuO${}_{4+\delta }$ has the same $\gamma$ value as other cuprates at $p\simeq 0.3$, this implies that $\gamma$ in HgBa${}_2$CuO${}_{4+\delta }$ must peak between $p\simeq 0.09$ and $p \simeq 0.3$,namelyat(ornear)thecriticaldoping$p^\star$ where the pseudogap phase is expected to end ($p^{\star }\simeq 0.2$). Secondly, the high $\gamma$ value implies that the Fermi surface must consist of more than the single electron-like pocket detected by quantum oscillations in HgBa${}_2$CuO$_{4

• \delta}$at$p \simeq 0.09$,whoseeffectivemass$m^\star= 2.7\times m_0$yieldsonly$\gamma = 4.0$mJ\mathrm{/}K$^2$mol. This missing mass imposes a revision of the current scenario for how pseudogap and charge order respectively transform and reconstruct the Fermi surface of cuprates.