Magnetic fluctuations and specific heat in Na$_x$CoO$_2$ near a Lifshitz Fermi surface 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 $\Gamma$ point, in addition to the existing hole pocket with large $k_F$. The data show that near $x =x_c$, the temperature dependence of $C(T)/T$ at low $T$ gets stronger as $x$ approaches $x_c$ from below and then reverses the trend and changes sign at $x \geq x_c$. 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 $x \leq x_c$ and $x >x_c$, when the new pocket forms. We demonstrate that this explains the temperature dependence of $C(T)/T$. We show that at larger $x$ ($x > 0.65$) the system enters a magnetic quantum critical regime where $C(T)/T$ roughly scales as $\log T$. This behavior extends to progressively lower $T$ as $x$ increases towards a magnetic instability at $x \approx 0.75$.