Heat capacity for a two-dimensional linear fermion dispersion ($C_{+}=9g\zeta (3)A{k}_B^3{T}^2/(4\pi {\hslash }^2{v}^2)$)

For $g$ copies of a positive-energy branch $E=\hslash v|\mathbf{k}|$ in area $A$, the density of states is $D(E)=gAE/(2\pi {\hslash }^2{v}^2)$. At zero chemical potential, the thermal energy is $3g\zeta (3)A(k_{B}T{)}^3/(4\pi {\hslash }^2{v}^2)$, giving the displayed fixed-zero-chemical-potential heat capacity. If a symmetric filled negative-energy band also supplies holes, its excitation energy contributes an equal amount. Counting cones, spin species and holes explicitly is necessary before assigning an absolute numerical coefficient.