Anomalous Thermodynamics of Coulomb-Interacting Massless Dirac Fermions in Two Spatial Dimensions

It is argued that the specific heat of $N$ massless Dirac fermions in two spatial dimensions interacting with $1/r$ Coulomb interactions is suppressed logarithmically relative to its noninteracting counterpart. The (dimensionless) coefficient of the logarithm is calculated in a closed form in the leading order in large $N$ expansion, but to all orders in the effective fine structure constant, ${\alpha }_F$, a procedure which takes into account finite temperature screening. This effect is expected to occur in a single-layer graphene embedded in a dielectric medium. Its dependence on the dielectric constant is calculated analytically.

Figure 1

The coefficient $\eta =\frac{1}{N}[\frac{16}{{\pi }^2}{\lambda }^{2}g(\lambda )-\frac{4}{\pi \lambda }]$ of the logarithmic increase of the free energy in Eq. (2) versus $\lambda =N{e}_{*}^2/16=N\frac{\pi }{8}\frac{e^2}{ϵ\hslash v_F}$. The closed form expression for $g$ is given in Eq. (12). For small $\lambda$, $\eta =2\lambda /(\pi N)$, while for $\lambda \to \infty$, $\eta =8/({\pi }^{2}N)$.

Figure 2

The suppression of the specific heat (per Dirac cone) relative to the noninteracting result $c_V^{(0)}=\frac{9\zeta (3)}{2\pi }{k}_B^3{T}^2/(\hslash v_F{)}^2$ (dashed line) versus $T/T_{\mathrm{UV}}$. This plot is for $N=4$, $v_F={10}^{6}m/s$, $ϵ=1$, which gives $\eta \approx 0.14$.