Magnetic fluctuations and specific heat in Na x CoO 2 near a Lifshitz Fermi surface topological transition Sergey Slizovskiy, 1 Andrey V. Chubukov, 2 and Joseph J. Betouras 1 1 Department of Physics, Loughborough University, Loughborough LE11 3TU, UK 2 Department of Physics,University of Wisconsin-Madison, Madison, WI 53706, USA We analyze the temperature and doping dependence of the specific heat C(T ) in NaxCoO2. This material was conjectured to undergo a Lifshitz -type topological transition at x = xc =0.62, in which a new electron Fermi pocket emerges at the Γ point, in addition to the existing hole pocket with large kF . The data show that near x = xc, the temperature dependence of C(T )/T at low T gets stronger as x approaches xc from below and then reverses the trend and changes sign at x ≥ xc. We argue that this behavior can be quantitatively explained within the spin-fluctuation theory. We show that magnetic fluctuations are enhanced near xc at momenta around kF and their dynamics changes between x ≤ xc and x>xc, 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 ≈ 0.75. Introduction The layered cobaltates Na x CoO 2 have been the subject of intense studies in recent years due to their very rich phase diagram and as- sociated rich physics 1–7 . Their structure is similar to that of copper oxides and consists of alterna- tively stacked layers of CoO 2 separated by sodium ions. The Co atoms form a triangular lattice 8 . The hydrated compound Na x CoO 2 :yH 2 O with x ∼ 0.3 shows superconductivity 9 , most likely of electronic origin. The anhydrated parent compound Na x CoO 2 exhibits low resistivity and thermal conductivity and high thermopower 1,2 for 0.5 <x< 0.9 and magnetic order for 0.75 <x< 0.9 (Refs.6,7,10,11). In the paramagnetic phase Na x CoO 2 shows a conventional metallic behavior at x ≤ 0.6 and at larger x displays strong temperature dependence of both spin suscep- tibility and specific heat down to very low T . This change of behavior has been attributed 12 to a pu- tative Lifshitz-type topological transition 13 (LTT) at x c ≈ 0.62, in which a small three-dimensional (3D) electron Fermi pocket appears around k = 0, in addition to the already existing quasi-2D hole pocket with large k F 1 (Ref.14), see Fig. 1. Although the small pocket has not yet been observed directly, ARPES measurements at smaller x did find a lo- cal minimum in the quasiparticle dispersion at the Γ point 15 . Similar topological transitions have been either observed or proposed for several solid state [16–23] and cold atom systems [24], and the under- standing of the role played by the interactions near the LTT transition is of rather general interest to condensed matter and cold atoms communities. The subject of this paper is the analysis of in- teraction contributions to the specific heat C (T ) in Na x CoO 2 at around the critical x c for LTT. The ex- perimental data 12 , show (see Figs. 3 and 4) that for doping near x c , the temperature dependence of C (T )/T is more complex than the C (T )/T = γ 1 + γ 3 T 2 + O(T 4 ) expected in an ordinary Fermi liquid (FL). The FL behavior itself is not broken in FIG. 1: The lattice fermionic dispersion ǫ(k) at kx =0 (in units of t1 ≈ 0.1eV ). See 25 for the values of the other hopping integrals. Note that the dispersion is approxi- mately rotationally invariant in the kx − ky plane and is quite shallow: the depth of the local minimum is around 20 meV. the sense that γ 1 remains finite. However the T de- pendence at x = x c is stronger than T 2 , as evidenced by the fact that the fits of the data on C (T )/T to γ 1 + γ 3 T 2 behavior 12 in finite intervals around dif- ferent T yield larger γ 3 as T goes down (see Ref.36). This does not allow one to interpret γ 1 directly as a density of states, and the full computation is needed to compare the data with the theory. For doping lev- els 0.65 <x< 0.75 the data show 3 that, to a good approximation, C (T )/T ∝ log T in a wide range of temperatures T ∼ 1 – 10 K, see Fig. 4a. This loga- rithmic temperature dependence progressively spans over larger temperature range as x approaches 0.75, where a magnetic order develops (Refs.[6,7,10,11]). Some qualitative features of the experimental data of C (T ) at x ∼ x c are reproduced by the free- fermion formula for specific heat, with the quasi- particle dispersion taken from first-principle calcu- lations (Fig. 2a). In particular, γ 1 increases and γ 3 passes through a maximum around x =0.62, see Fig. 3b,c . However, the magnitudes of γ 1 and γ 3 are much smaller than in the data and the maxi- arXiv:1409.0408v3 [cond-mat.str-el] 13 Jan 2015