We consider a theory of massless reduced quantum electrodynamics (RQED$_{d_\gamma,d_e}$), e.g., a quantum field theory where the U(1) gauge field lives in $d_\gamma$-spacetime dimensions while the fermionic field lives in a reduced spacetime of $d_e$ dimensions ($d_e \leqslant d_\gamma$). In the case where $d_\gamma=4$ such RQEDs are renormalizable while they are super-renormalizable for $d_\gamma <4$. The 2-loop electromagnetic current correlation function is computed exactly for a general RQED$_{d_\gamma,d_e}$. Focusing on RQED$_{4,3}$, the corresponding $\beta$-function is shown to vanish which implies the scale invariance of the theory. Interaction correction to the 1-loop vacuum polarization, $\Pi_1$, of RQED$_{4,3}$ is found to be: $\Pi = \Pi_1 (1 + 0.056 \al)$ where $\al$ is the fine structure constant. The scaling dimension of the fermion field is computed at 1-loop and is shown to be anomalous for RQED$_{4,3}$.Comment: (v2) Accepted for publication in PRD. Conclusion and references added (some / referee's comments). No change in results. 8 pages, 3 figures. (v1) LaTeX file with feynMF package. 8 pages, no figur

We study the effect of impurities on the ground state and the low-temperature Ohmic dc transport in a one-dimensional chain and quasi-one-dimensional systems of many parallel chains. We assume that strong interactions impose a short-range periodicicity of the electron positions. The longrange order of such an electron crystal (or equivalently, a 4kF charge-density wave) is destroyed by impurities, which act as strong pinning centers. We show that a three-dimensional array of chains behaves differently at large and at small impurity concentrations N . At large N , impurities divide the chains into metallic rods. Additions or removal of electrons from such rods correspond to charge excitations whose density of states exhibits a quadratic Coulomb gap. At low temperatures the conductivity is due to the variable-range hopping of electrons between the rods. It obeys the Efros-Shklovskii (ES) law − ln σ ∼ (TES/T ) 1/2 . TES decreases as N decreases, which leads to an exponential growth of σ. When N is small, the metallic-rod (also known as "interrupted-strand") picture of the ground state survives only in the form of rare clusters of atypically short rods. They are the source of low-energy charge excitations. In the bulk of the crystal the charge excitations are gapped and the electron crystal is pinned collectively. A strongly anisotropic screening of the Coulomb potential produces an unconventional linear in energy Coulomb gap and a new law of the variable-range hopping conductivity − ln σ ∼ (T1/T ) 2/5 . The parameter T1 remains constant over a finite range of impurity concentrations. At smaller N the 2/5-law is replaced by the Mott law, − ln σ ∼ (TM /T ) 1/4 . In the Mott regime the conductivity gets suppressed as N goes down. Thus, the overall dependence of σ on N is nonmonotonic. In the case of a single chain, the metallic-rod picture applies at all N . The low-temperature conductivity obeys the ES law, with log-corrections, and decreases exponentially with N . Our theory provides a qualitative explanation for the transport properties of organic charge-density wave compounds of TCNQ family.

We compute the two-loop fermion self-energy in massless reduced quantum electrodynamics for an arbitrary gauge using the method of integration by parts. Focusing on the limit where the photon field is four-dimensional, our formula involves only recursively one-loop integrals and can therefore be evaluated exactly. From this formula, we deduce the anomalous scaling dimension of the fermion field as well as the renormalized fermion propagator up to two loops. The results are then applied to the ultra-relativistic limit of graphene and compared with similar results obtained for four-dimensional and three-dimensional quantum electrodynamics.