L\'evy flights due to anisotropic disorder in graphene

We study transport properties of graphene with anisotropically distributed on-site impurities (adatoms) that are randomly placed on every third line drawn along carbon bonds. We show that stripe states characterized by strongly suppressed back-scattering are formed in this model in the direction of the lines. The system reveals L\'evy-flight transport in stripe direction such that the corresponding conductivity increases as the square root of the system length. Thus, adding this type of disorder to clean graphene near the Dirac point strongly enhances the conductivity, which is in stark contrast with a fully random distribution of on-site impurities which leads to Anderson localization. The effect is demonstrated both by numerical simulations using the Kwant code and by an analytical theory based on the self-consistent $T$-matrix approximation.Comment: 11 pages, 6 figure

Proximity effect in the presence of Coulomb interaction and magnetic field

We consider a small metallic grain coupled to a superconductor by a tunnel contact. We study the interplay between proximity and charging effects in the presence of the external magnetic field. Employing the adiabatic approximation we develop a self-consistent theory valid for an arbitrary ratio of proximity and Coulomb strength. The magnetic field suppresses the proximity-induced minigap in an unusual way. We find the phase diagram of the grain in the charging energy - magnetic field plane. Two distinct states exist with different values and magnetic field dependences of the minigap. The first-order phase transition occurs between these two minigapped states. The transition to the gapless state may occur by the first- or second-order mechanism depending on the charging energy. We also calculate the tunneling density of states in the grain. The energy dependence of this quantity demonstrates two different gaps corresponding to the Coulomb and proximity effects. These gaps may be separated in sufficiently high magnetic field.Comment: 11 pages (including 8 EPS figures). Version 3: extended. Final version as published in PR

Anomalous Hall effect in disordered Weyl semimetals

We study the anomalous Hall effect in a disordered Weyl semimetal. While the intrinsic contribution is expressed solely in terms of Berry curvature, the extrinsic contribution is given by a combination of the skew scattering and side jump terms. For the model of small size impurities, we are able to express the skew scattering contribution in terms of scattering phase shifts. We identify the regime in which the skew scattering contribution dominates the side-jump contribution: the impurities are either strong or resonant, and at dilute concentration. In this regime, the Hall resistivity $\rho_{xy}$ is expressed in terms of two scattering phases, analogous to the s-wave scattering phase in a non-topological metal. We compute the dependence of $\rho_{xy}$ on the chemical potential, and show that $\rho_{xy}$ scales with temperature as $T^2$ in low temperatures and as $T^{3/2}$ in the high temperature limit

Quantum criticality and minimal conductivity in graphene with long-range disorder

We consider the conductivity $\sigma_{xx}$ of graphene with negligible intervalley scattering at half filling. We derive the effective field theory, which, for the case of a potential disorder, is a symplectic-class $\sigma$-model including a topological term with $\theta=\pi$. As a consequence, the system is at a quantum critical point with a universal value of the conductivity of the order of $e^2/h$. When the effective time reversal symmetry is broken, the symmetry class becomes unitary, and $\sigma_{xx}$ acquires the value characteristic for the quantum Hall transition.Comment: 4 pages, 1 figur

Heat transport in Weyl semimetals in the hydrodynamic regime

We study heat transport in a Weyl semimetal with broken time-reversal symmetry in the hydrodynamic regime. At the neutrality point, the longitudinal heat conductivity is governed by the momentum relaxation (elastic) time, while longitudinal electric conductivity is controlled by the inelastic scattering time. In the hydrodynamic regime this leads to a large longitudinal Lorenz ratio. As the chemical potential is tuned away from the neutrality point, the longitudinal Lorenz ratio decreases because of suppression of the heat conductivity by the Seebeck effect. The Seebeck effect (thermopower) and the open circuit heat conductivity are intertwined with the electric conductivity. The magnitude of Seebeck tensor is parametrically enhanced, compared to the non-interacting model, in a wide parameter range. While the longitudinal component of Seebeck response decreases with increasing electric anomalous Hall conductivity $\sigma_{xy}$, the transverse component depends on $\sigma_{xy}$ in a non-monotonous way. Via its effect on the Seebeck response, large $\sigma_{xy}$ enhances the longitudinal Lorenz ratio at a finite chemical potential. At the neutrality point, the transverse heat conductivity is determined by the Wiedemann-Franz law. Increasing the distance from the neutrality point, the transverse heat conductivity is enhanced by the transverse Seebeck effect and follows its non-monotonous dependence on $\sigma_{xy}$