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 ($p^starsimeq 0.2$). 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 $m^star= 2.7times m_0$ 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.