We investigate the local density of states and Friedel oscillation in graphene around a well localized impurity in Born approximation. In our analytical calculations Green's function technique has been used taking into account both the localized atomic wavefunctions in a tight-binding scheme and the corresponding symmetries of the lattice. As a result we obtained long wavelength oscillations in the density of electrons with long range behavior proportional to the inverse square of the distance from the impurity. These leading oscillations are out of phase on nearby lattice sites (in fact for an extended defect they cancel each other within one unit cell), therefore a probe with resolution worse than a few unit cells will experience only the next to leading inverse cube decay of density oscillations even for a short range scatterer. PACS numbers: 81.05.ue, 73.22.Pr, 73.22.Dj, 74.55.+v Ever since the first production of atomically thin carbon films 1 , graphene continues to fascinate physicists of almost all walks of life. This simple two dimensional system of carbon atoms arranged in a honeycomb lattice provides us with a number of interesting properties due mainly to the massless Dirac nature of the dispersion relation of its electrons 2 . Undoped graphene has its Fermi energy at the tip of the Dirac cones and behaves as a zero gap semiconductor. Applying appropriate gate voltage leads to electron or hole pockets and metallic behavior with Fermi wavenumber k F typically much smaller than the size of the Brillouin zone. Exciting potential applications of graphene include for example carbon based planar electronic circuitry with the possibility of electrically reconfigurable wiring 3 , exploiting the guiding effect of graphene p-n junctions with negative refractive index 4 . On the theoretical side perhaps the simplest quantity to be considered is the change in the local density of states (LDOS) and the Friedel oscillation (FO) in the excess charge density due to a well localized impurity. Results of these considerations have implications for the LDOS in disordered graphene 5 , for the interaction between adatoms in graphene 6 , or in case of magnetic adatoms 7 for the corresponding RKKY interaction 8 .Early theoretical work on the LDOS and the resulting FO around an impurity in graphene 9,10 predicted long wavelength (2k F ) oscillations in the charge density, but with envelope decaying like r −3 at distance r from the impurity. This is in contrast to the r −2 decay in a degenerate nonrelativistic two dimensional Fermi gas, and suppressed backscattering of chiral graphene electrons residing around the Fermi circle of the Dirac cone was offered as an explanation. However, graphene has two inequivalent Dirac cones (valleys) in the Brillouin zone, and intervalley scattering by the impurity may lead to short wavelength oscillations on the order of a few lattice constants as well. Indeed, a scanning tunneling microscopy (STM) study 11 of epitaxial graphene revealed two different length scales around defects. Subsequ...

We analyze the probability distribution function (PDF) of work done on a Luttinger liquid for an arbitrary finite duration interaction quench and show that it can be described in terms of a generalized Gibbs ensemble. We construct the corresponding density matrix with explicit intermode correlations, and determine the duration and interaction dependence of the probability of an adiabatic transition and the PDF of nonadiabatic processes. In the thermodynamic limit, the PDF of work exhibits a non-Gaussian maximum around the excess heat, carrying almost all the spectral weight. In contrast, in the small system limit most spectral weight is carried by a delta peak at the energy of the adiabatic process, and an oscillating PDF with dips at energies commensurate to the quench duration and with an exponential envelope develops. Relevance to cold atom experiments is also discussed.

We investigate optical transitions of non-interacting electron systems consisting of two symmetric energy bands touching each other at the Fermi energy (e.g. graphene at half filling). Optical conductivity is obtained using Kubo formula at zero temperature. We show that for particles whose pseudospin direction is determined solely by the direction of their momentum, the optical conductivity has power law frequency dependence with the exponent (d − 2)/z where d is the dimension of the system and z is the dynamical exponent. According to our result two-dimensional systems with the above pseudospin characteristics always exhibit frequency-independent optical conductivity.

We study the heating dynamics of a Luttinger liquid, upon suddenly coupling it to a dissipative environment. Within the Lindblad equation, the environment couples to local currents and heats the quantum liquid up to infinite temperatures. The fermionic single particle density matrix, and many other correlators, retain the initial Luttinger liquid correlations in space but decay exponentially in time with a rate that depends on the strength of the interaction. The spectrum of the time evolved density matrix is gapped, which collapses gradually as − ln(t). The von Neumann entropy crosses over from the early time −t ln(t) behaviour to a ln(t) growth for late times. The early time dynamics is captured numerically by performing simulations on spinless interacting fermions, using several numerically exact methods. Our results establish the validity of bosonization for the early time dynamics of 1D lattice models, and could be observed experimentally in bosonic Luttinger liquids.