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 $T_{\rm c} = 72$ K at temperatures down to $2$ K in magnetic fields up to $35$ T, a field large enough to suppress superconductivity at that doping ($p \simeq 0.09$). In the normal state at $H = 35$ T, a residual linear term of magnitude $\gamma = 12 \pm 2$ mJ/K$^2$mol is observed in $C/T$ 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/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 $p \simeq 0.3$, namely at (or near) the critical doping $p^\star$ 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 $p \simeq 0.09$, whose effective mass $m^\star= 2.7\times m_0$ yields only $\gamma = 4.0$ mJ/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.