Conductivity of pure graphene: Theoretical approach using the polarization tensor

We obtain analytic expressions for the conductivity of pristine (pure) graphene in the framework of the Dirac model using the polarization tensor in (2+1)-dimensions defined along the real frequency axis. It is found that at both zero and nonzero temperature $T$ the in-plane and out-of-plane conductivities of graphene are equal to each other with a high precision and essentially do not depend on the wave vector. At $T=0$ the conductivity of graphene is real and equal to $\sigma_0=e^2/(4\hbar)$ up to small nonlocal corrections in accordance with many authors. At some fixed $T\neq 0$ the real part of the conductivity varies between zero at low frequencies $\omega$ and $\sigma_0$ for optical $\omega$. If $\omega$ is fixed, the conductivity varies between $\sigma_0$ at low $T$ and zero at high $T$. The imaginary part of the conductivity of graphene is shown to depend on the ratio of $\omega$ to $T$. In accordance to the obtained asymptotic expressions, at fixed $T$ it varies from infinity at $\omega=0$ to a negative minimum value reached at some $\omega$, and then approaches to zero with further increase of $\omega$. At fixed $\omega$ the imaginary part of the conductivity varies from zero at $T=0$, reaches a negative minimum at some $T$ and then goes to infinity together with $T$. The numerical computations of both the real and imaginary parts of the conductivity are performed. The above results are obtained in the framework of quantum electrodynamics at nonzero temperature and can be generalized for graphene samples with nonzero mass gap parameter and chemical potential.