Two-parameter scaling theory of transport near a spectral node

We investigate the finite-size scaling behavior of the conductivity in a two-dimensional Dirac electron gas within a chiral sigma model. Based on the fact that the conductivity is a function of system size times scattering rate, we obtain a two-parameter scaling flow toward a finite fixed point. The latter is the minimal conductivity of the infinite system. Depending on boundary conditions, we also observe unstable fixed points with conductivities much larger than the experimentally observed values, which may account for results found in some numerical simulations. By including a spectral gap we extend our scaling approach to describe a metal-insulator transition.Comment: 4.5 pages, 4 figures, published versio

Renormalized transport properties of randomly gapped 2D Dirac fermions

We investigate the scaling properties of the recently acquired fermionic non--linear $\sigma$--model which controls gapless diffusive modes in a two--dimensional disordered system of Dirac electrons beyond charge neutrality. The transport on large scales is governed by a novel renormalizable nonlocal field theory. For zero mean random gap, it is characterized by the absence of a dynamic gap generation and a scale invariant diffusion coefficient. The $\beta$ function of the DC conductivity, computed for this model, is in perfect agreement with numerical results obtained previously.Comment: Version published with minor change

Quantum Hall effect induced by electron-phonon interaction

When phonons couple to fermions in 2D semimetals, the interaction may turn the system into an insulator. There are several insulating phases in which the time reversal and the sublattice symmetries are spontaneously broken. Examples are many-body states commensurate to Haldane's staggered flux model or to lattice models with periodically modulated strain. We find that the effective field theories of these phases exhibit characteristic Chern-Simons terms, whose coefficients are related to the topological invariants of the microscopic model. This implies that the corresponding quantized Hall conductivities characterize these insulating states.Comment: Accepted for publishing with Annals of Physics on April 30th, 202

Perturbative analysis of the conductivity in disordered monolayer and bilayer graphene

The DC conductivity of monolayer and bilayer graphene is studied perturbatively for different types of disorder. In the case of monolayer, an exact cancellation of logarithmic divergences occurs for all disorder types. The total conductivity correction for a random vector potential is zero, while for a random scalar potential and a random gap it acquires finite corrections. We identify the diagrams which are responsible for these corrections and extrapolate the finite contributions to higher orders which gives us general expressions for the conductivity of weakly disordered monolayer graphene. In the case of bilayer graphene, a cancellation of all contributions for all types of disorder takes place. Thus, the minimal conductivity of bilayer graphene turns out to be very robust against disorder.Comment: 4 pages, 2 figures + supplementary material. Final version as published with PR

Effect of Coulomb interaction on the gap in monolayer and bilayer graphene

We study effects of a repulsive Coulomb interaction on the spectral gap in monolayer and bilayer graphene in the vicinity of the charge neutrality point by employing the functional renormalization-group technique. In both cases Coulomb interaction supports the gap once it is open. For monolayer graphene we correctly reproduce results obtained previously by several authors, e.g., an apparent logarithmic divergence of the Fermi velocity and the gap as well as a fixed point corresponding to a quantum phase transition at infinitely large Coulomb interaction. On the other hand, we show that the gap introduces an additional length scale at which renormalization flow of diverging quantities saturates. An analogous analysis is also performed for bilayer graphene with similar results. We find an additional fixed point in the gapless regime with linear spectrum corresponding to the vanishing electronic band mass. This fixed point is unstable with respect to gap fluctuations and can not be reached as soon as the gap is opened. This preserves the quadratic scaling of the spectrum and finite electronic band mass.Comment: 6 pages, 5 figures, final version to appear at PR

Spontaneous mass generation due to phonons in a two-dimensional Dirac fermion system

Fermions with one and two Dirac nodes are coupled to in-plane phonons to study a spontaneous transition into the Hall insulating phase. At sufficiently strong electron-phonon interaction a gap appears in the spectrum of fermions, signaling a transition into a phase with spontaneously broken parity and time-reversal symmetry. The structure of elementary excitations above the gap in the corresponding phase reveals the presence of scale invariant parity breaking terms which resemble Chern-Simons excitations. Evaluating the Kubo formula for both models we find quantized Hall plateaux in each case, with conductance of binodal model exactly twice as large as of the mononodal model

Linear response peculiarity of a two--dimensional Dirac electron gas at weak scattering

The conductivity of an electron gas can be alternatively calculated either from the current--current or from the density--density correlation function. Here, we compare these two frequently used formulations of the Kubo formula for the two--dimensional Dirac electron gas by direct evaluations for several special cases. Assuming the presence of weak disorder we investigate perturbatively both formulas at and away from the Dirac point. While to zeroth order in the disorder amplitude both formulations give identical results, with some very strong assumptions though, they show significant discrepancies already in first order. At half filling we evaluate all second order diagrams. Virtually none of the topologically identical diagrams yield the same corrections for both formulations. We conclude that a direct comparison of conductivities of disordered system calculated in both formulas is not possible.Comment: Published versio