Two-dimensional massless Dirac fermions exhibit Dirac cones, which are classified into three types: type I, type II, and type III. In both type-I and type-II cones, the energy dispersion is linear in all momentum directions. Type-I cones are characterized by a nonovertilted structure, where the Dirac point serves as a local minimum (maximum) for the upper (lower) band. In contrast, type-II cones exhibit overtilted dispersions, leading to the coexistence of electron and hole pockets. At the critical tilt, the linear energy dispersion vanishes in one momentum direction, corresponding to a type-III Dirac cone. We further define a special case, termed the “narrow-sense” type-III cone, where not only the linear term but also quadratic and higher-order terms vanish, resulting in a completely flat dispersion along one direction. In this work, we numerically investigate the temperature ($T$) dependence of the electronic specific heat ($C$), as the Dirac cone is continuously tilted from type I to narrow-sense type III. A model with particle-hole symmetry is employed to ensure that the chemical potential ($\mu$) remains temperature independent. Our results reveal a notable crossover in $C$ near narrow-sense type III, where $C$ changes from $C\propto T^2$ below the crossover temperature ($T_{\mathrm{co}}$) to $C\propto T^{\frac{1}{2}}$ above $T_{\mathrm{co}}$. This crossover is attributed to the energy-dependent structure of the density of states. The present findings suggest a feasible approach for experimentally probing the degree of Dirac cone tilting near the narrow-sense type-III limit.

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• Specific heat
• Topological materials
• 2-dimensional systems
• Graphene