Entropy spikes as a signature of Lifshitz transitions in the Dirac materials

We demonstrate theoretically that the characteristic feature of a 2D system undergoing $N$ consequent Lifshitz topological transitions is the occurrence of spikes of entropy per particle $s$ of a magnitude $pm ln 2/(J-1/2)$ with $2 leq J leq N$ at low temperatures. We derive a general expression for $s$ as a function of chemical potential, temperature and gap magnitude for the gapped Dirac materials. Inside the smallest gap, the dependence of $s$ on the chemical potential exhibits a dip-and-peak structure in the temperature vicinity of the Dirac point. The spikes of the entropy per particles can be considered as a signature of the Dirac materials. These distinctive characteristics of gapped Dirac materials can be detected in transport experiments where the temperature is modulated in gated structures.