Magnetic fluctuations and specific heat in NaxCoO2 near a Lifshitz topological transition

We analyze the temperature and doping dependence of the specific heat $C(T)$ in Na${}_x$CoO${}_2$. This material was conjectured to undergo a Lifshitz -type topological transition at $x=x_c=0.62$, in which a new electron Fermi pocket emerges at the $\mathrm{\Gamma }$ point, in addition to the existing hole pocket with large $k_F$. The data show that near $x=x_c$, the temperature dependence of $C(T)\mathrm{/}T$ at low $T$ gets stronger as $x$ approaches $x_c$ from below and then reverses the trend and changes sign at $x\ge x_c$. We argue that this behavior can be quantitatively explained within the spin-fluctuation theory. We show that magnetic fluctuations are enhanced near $x_c$ at momenta around $k_F$ and their dynamics changes between $x\le x_c$ and $x>x_c$, when the new pocket forms. We demonstrate that this explains the temperature dependence of $C(T)\mathrm{/}T$. We show that at larger $x$ ($x>0.65$) the system enters a magnetic quantum critical regime where $C(T)\mathrm{/}T$ roughly scales as $\log T$. This behavior extends to progressively lower $T$ as $x$ increases towards a magnetic instability at $x\approx 0.75$.