The process of coherent creation of particle -hole excitations by an electric field in graphene is quantitatively described. We calculate the evolution of current density, number of pairs and energy after switching on the electric field. In particular, it leads to a dynamical visualization of the universal finite resistivity without dissipation in pure graphene. We show that the DC conductivity of pure graphene is . This value coincides with the AC conductivity calculated and measured recently at optical frequencies. The effect of temperature and random chemical potential (charge puddles) are considered and explain the recent experiment on suspended graphene. A possibility of Bloch oscillations is discussed within the tight binding model.

The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian can be extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model when a transition is made to an associated manifold for which the geodesics coincide with the orbits of the Hamiltonian potential model. We therefore find a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this associated manifold generates a dynamical curvature, resulting in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions giving, as a particular illustation, detailed results for a potential obtained from a fifth order expansion of a Toda lattice Hamiltonian.

The process of coherent creation of particle -hole excitations by an electric field in graphene is quantitatively described beyond linear response. We calculate the evolution of current density, number of pairs and energy in ballistic regime for electric field E using the tight binding model. While for ballistic flight times smaller than t nl ∝ E −1/2 current is linear in E and independent of time, for larger ballistic times the current increases after t nl as J ∝ E 3/2 t and finally at yet larger times (t > tB ∝ E −1 ) Bloch oscillations set in. It is shown that the number of pairs follows the 2D generalization of the Schwinger's creation rate n ∝ E 3/2 only on certain time segments with a prefactor different from that obtained using the asymptotic formula.

The long standing controversy concerning the effect of electron -electron interaction on the electrical conductivity of an ideal graphene sheet is settled. Performing the calculation directly in the tight binding approach without the usual prior reduction to the massless Dirac (Weyl) theory, it is found that, to leading order in the interaction strength α = e 2 / v0, the DC conductivity σ/σ0 = 1 + Cα + O α 2 is significantly enhanced with respect to the independent-electrons result σ0, i.e. with the value C = 0.26. The ambiguity characterizing the various existing approaches is nontrivial and related to the chiral anomaly in the system. In order to separate the energy scales in a model with massless fermions, contributions from regions of the Brillouin zone away from the Dirac points have to be accounted for. Experimental consequences of the relatively strong interaction effect are briefly discussed.

The process of coherent creation of particle -hole excitations by an electric field in graphene is quantitatively described using a dynamic "first quantized" approach. We calculate the evolution of current density, number of pairs and energy in ballistic regime using the tight binding model. The series in electric field strength E up to third order in both DC and AC are calculated. We show how the physics far from the two Dirac points enters various physical quantities in linear response and how it is related to the chiral anomaly. The third harmonic generation and the imaginary part of conductivity are obtained. It is shown that at certain time scale t nl ∝ E −1/2 the physical behaviour dramatically changes and the perturbation theory breaks down. Beyond the linear response physics is explored using an exact solution of the first quantized equations. While for small electric fields the I-V curve is linear characterized by the universal minimal resistivity σ = π/2(e 2 /h), at t > t nl the conductivity grows fast. The copious pair creation (with rate E 3/2 ), analogous to Schwinger's electron -positron pair creation from vacuum in QED, leads to creation of the electron -hole plasma at ballistic times of order t nl . This process is terminated by a relaxational recombination.