Observable precursor of topological phase transition: temperature-dependent electronic specific heat in two-dimensional Dirac fermions

Dirac points in two-dimensional massless Dirac fermions are topologically protected. Although single Dirac point cannot disappear solely, a pair of two Dirac points annihilates after merging at a time-reversal invariant momentum (TRIM). This process triggers a topological phase transition. In this paper, we numerically calculate the electronic specific heat ($C$) of the systems with a honeycomb lattice and $\alpha$-(BEDT-TTF)$_2$I$_3$ in the case that two Dirac points are moving and merging by changing the ratio of the magnitude between the transfer integrals, which can be controlled by uniaxial pressure for example. When two Dirac points are close to but not at TRIM, the temperature dependence exhibits a crossover from $C\propto T^{2.0}$ (expected for separated Dirac points) at low temperatures ($T\leq T_{\rm co}$) to $C\propto T^{1.5}$ (expected in the case of the merged Dirac points) at high temperatures ($T\geq T_{\rm co}$). Here, $T_{\rm co}$ denotes the crossover temperature, which is determined by the potential barrier magnitude between two Dirac points. Our findings demonstrate that the precursor of the topological phase transition is observed through the temperature-dependence of the electronic specific heat.