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The properties of zero modes in particle-hole symmetric systems are analyzed in the presence of strong random scattering by a disordered environment. The study is based on the calculation of the time-averaged density distribution on a lattice. In particular, a flat distribution is found for strong random scattering. This result is compared with a decaying distribution for weak random scattering by an analysis of the scattering paths. In the calculation we consider the invariant measure of the average two-particle Green's function, which is related to lattice-covering self-avoiding (LCSA) strings. In particular, strong scattering is associated with LCSA loops, whereas weaker scattering is associated with open LCSA strings. Our results are a generalization of the delocalized state observed at the band center of a one-dimensional tight-binding model with random hopping by Dyson in 1953., Comment: 12 pages, 3 figures

A finite photonic lattice with two bands and a random gap is considered. Using a two-dimensional Dirac equation, the effect of a random sign of the Dirac mass is studied numerically. The edge state at the sample boundary has a strong influence on the electromagnetic field and its polarization inside the sample. The creation of edge states through a randomly fluctuating sign of the Dirac mass defeats Anderson localization and allows the electromagnetic field to distribute over the entire sample. The width of the distribution increases with an increasing gap due to increasing sharpness of the edge states. These results are compared with those of a random one-band Helmholtz equation. In contrast to the Dirac model, the one-band model displays a clear signature of Anderson localization., Comment: 12 pages, 12 figures

The average density of states of two- and three-dimensional Weyl fermions is studied in the self-consistent Born approximation (SCBA) and its corrections. The latter have been organized in terms of a 1/N expansion. It turns out that an expansion in terms of the disorder strength is not applicable, as previously mentioned by other authors. Nevertheless, the 1/N expansion provides a justification of the SCBA as the large N limit of Weyl fermions., Comment: 12 pages, 4 figures, Phys. Rev. B (2017)

The average density of states in a disordered three-dimensional Weyl system is discussed in the case of a continuous distribution of random scattering. Our result clearly indicate that the average density of states does not vanish, reflecting the absence of a critical point for a metal-insulator transition. This calculation supports recent suggestions of an avoided quantum critical point in the disordered three-dimensional Weyl semimetal. However, the effective density of states can be very small such that the saddle-approximation with a vanishing density of states might be valid for practical cases., Comment: 5 pages, 2 figures, minor changes, additional supplement

In the presence of strong random scattering the behavior of particles with degenerate spectra is quite different from Anderson localization of particles in a single band: it creates geometric states rather than confining the particles to an area of the size of the localization length. These states are subject to a Fokker-Planck dynamics with universal drift velocity and disorder dependent diffusion coefficient. This behavior has some similarity with the unidirectionally propagating edge states in quantum Hall systems., Comment: 5 pages, 3 figures

We calculate the transport properties of three-dimensional Weyl fermions in a disordered environment. The resulting conductivity depends only on the Fermi energy and the scattering rate. First we study the conductivity at the spectral node for a fixed scattering rate and obtain a continuous transition from an insulator at weak disorder to a metal at stronger disorder. In the self-consistent Born approximation the scattering rate depends on the Fermi energy. Then it is crucial that the limits of the conductivity for a vanishing Fermi energy and a vanishing scattering rate do not commute. As a result, there is also metallic behavior in the phase with vanishing scattering rate and only a quantum critical point remains as an insulating state., Comment: 9 pages, 5 figures

Anderson localization is studied for two-dimensional Dirac fermions in the presence of strong random scattering. Averaging with respect to the latter leads to a graphical representation of the correlation function with entangled random walks and three-vertices which connect three different types of propagators. This approach indicates Anderson localization along a semi-infinite line, where the localization length is inversely proportional to the scattering rate., Comment: 8 pages, 2 figures

Transport properties of a two-band system with spectral nodes are studied in the presence of random scattering. Starting from a Grassmann functional integral, we derive a bosonic representation that is based on random phase fluctuations. Averaging leads to a graphical representation of the correlation function with entangled random walks and 4-vertices. In the strong scattering limit we derive a complex transition amplitude. For the example of two-dimensional Dirac fermions we obtain a localization length proportional to the inverse scattering rate., Comment: 10 pages, 3 figures

We study a weakly disordered 2D electron gas with two bands and a spectral node within the weak-localization approach and compare its results with those of Gaussian fluctuations around the self-consistent Born approximation. The appearance of diffusive modes depends on the type of disorder. In particular, we find for a random gap a diffusive mode only from ladder contributions, whereas for a random scalar potential the diffusive mode is created by ladder and by maximally crossed contributions. The ladder (maximally crossed) contributions correspond to fermionic (bosonic) Gaussian fluctuations. We calculate the conductivity corrections from the density--density Kubo formula and find a good agreement with the experimentally observed V-shape conductivity of graphene., Comment: 10 pages, 2 figures

We study the properties of the spinor wavefunction in a strongly disordered environment on a two-dimensional lattice. By employing a transfer-matrix calculation we find that there is a transition from delocalized to localized states at a critical value of the disorder strength. We prove that there exists an Anderson localized phase with exponentially decaying correlations for sufficiently strong scattering. Our results indicate that suppressed backscattering is not sufficient to prevent Anderson localization of surface states in topological insulators., Comment: 8 pages, 2 figures, presentation simplified, one reference added

We use a random gap model to describe a metal-insulator transition in three-dimensional semiconductors due to doping and find a conventional phase transition, where the effective scattering rate is the order parameter. Spontaneous symmetry breaking results in metallic behavior, whereas the insulating regime is characterized by the absence of spontaneous symmetry breaking. The transition is continuous for the average conductivity with critical exponent equal to 1. Away from the critical point the exponent is roughly 0.6, which may explain experimental observations of a crossover of the exponent from 1 to 0.5 by going away from the critical point., Comment: 6 pages, 4 figures

We study the dynamics of non-interacting quantum particles with two bands in the presence of random scattering. The two bands are associated with a chiral symmetry. After breaking the latter by a potential, we still find that the quantum dynamics is controlled by a non-Abelian chiral symmetry. The possibility of spontaneous symmetry breaking is analyzed within a self-consistent approach, and the instability of a symmetric solution is discussed., Comment: 6 pages, no figures

We propose a lattice model for Dirac fermions which allows us to break the degeneracy of the node structure. In the presence of a random gap we analyze the scaling behavior of the localization length as a function of the system width within a numerical transfer-matrix approach. Depending on the strength of the randomness, there are different scaling regimes for weak, intermediate and strong disorder. These regimes are separated by transitions that are characterized by one-parameter scaling., Comment: 9 pages, 8 figures

We study a disordered 2D electron gas with a spectral node in a vicinity of the node. After identifying the fundamental dynamical symmetries of this system, the spontaneous breaking of the latter by a Grassmann field is studied within a nonlinear sigma model approach. This allows us to reduce the average two-particle Green's function to a diffusion propagator with a random diffusion coefficient. The latter has non-degenerate saddle points and is treated by the conventional self-consistent Born approximation. This leads to a renormalized chemical potential and a renormalized diffusion coefficient, where the DC conductivity increases linearly with the density of quasiparticles. Applied to the special case of Dirac fermions, our approach provides a comprehensive description of the minimal conductivity at the Dirac node as well as for the V-shape conductivity inside the bands., Comment: 13 pages, 4 figures, extended version

We study the scattering dynamics of an $n$-component spinor wavefunction in a random environment on a two-dimensional lattice. In the presence of particle-hole symmetry we find diffusion on large scales. The latter is described by a non-interacting Grassmann field, indicating a special kind of asymptotic freedom in $d=2$., Comment: 9 pages, no figures, extended version

We study the density of states in monolayer and bilayer graphene in the presence of a random potential that breaks sublattice symmetries. While a uniform symmetry-breaking potential opens a uniform gap, a random symmetry-breaking potential also creates tails in the density of states. The latter can close the gap again, preventing the system to become an insulator. However, for a sufficiently large gap the tails contain localized states with nonzero density of states. These localized states allow the system to conduct at nonzero temperature via variable-range hopping. This result is in agreement with recent experimental observations in graphane by Elias {\it et al.}., Comment: 16 pages, 7 figures

In this paper we study the effect of a fluctuating gap in mono- and bilayer graphene, created by a random staggered potential. We identify a continuous symmetry for the two-particle Green's function which is spontaneously broken in the average two-particle Green's function and leads to a massless fermion mode. Within a loop expansion it is shown that the massless mode is dominated on large scales by small loops. This result indicates diffusion of electrons. Although the diffusion mechanism is the same in mono- and in bilayer graphene, the amount of scattering is much stronger in the latter. Physical quantities at the neutrality point, such as the density of states, the diffusion coefficient and the conductivity, are determined by the one-particle scattering rate. All these quantities vanish at a critical value of the average staggered potential, signaling a continuous transition to an insulating behavior., Comment: 16 pages, 2 figures, extended version

A mixture of two fermionic species with different masses is studied in an optical lattice. The heavy fermions are subject only to thermal fluctuations, the light fermions also to quantum fluctuations. We derive the Ising-like distribution for the heavy atoms and study the localization properties of the light fermions numerically by a transfer-matrix method. In a two-dimensional system one-parameter scaling of the localization length is found with a transition from delocalized states at low temperatures to localized states at high temperature. The critical exponent of the localization length is $\nu\approx 0.88$., Comment: 5 pages, 4 figures

The minimal conductivity of graphene is a quantity measured in the DC limit. It is shown, using the Kubo formula, that the actual value of the minimal conductivity is sensitive to the order in which certain limits are taken. If the DC limit is taken before the integration over energies is performed, the minimal conductivity of graphene is $4/\pi$ (in units of $e^2/h$) and it is $\pi/2$ in the reverse order. The value $\pi$ is obtained if weak disorder is included via a small frequency-dependent selfenergy. In the high-frequency limit the minimal conductivity approaches $\pi/2$ and drops to zero if the frequency exceeds the cut-off energy of the particles., Comment: 5 pages, 1 figure, extended version

A mixture of light and heavy spin-polarized fermionic atoms in an optical lattice is considered. Tunneling of the heavy atoms is neglected such that they are only subject to thermal fluctuations. This results in a complex interplay between light and heavy atoms caused by quantum tunneling of the light atoms. The distribution of the heavy atoms is studied. It can be described by an Ising-like distribution with a first-order transition from homogeneous to staggered order. The latter is caused by an effective nonlocal interaction due to quantum tunneling of the light atoms. A second-order transition is also possible between an ordered and a disordered phase of the heavy atoms.

We demonstrate that discrepancies between predicted low-energy quasiparticle properties in disordered 2D d-wave superconductors occur because of the unanticipated importance of disorder model details and normal-state particle-hole symmetry. This conclusion follows from numerically exact evaluations of the quasiparticle density-of-states predicted by the Bogoliubov-deGennes (BdG) mean field equations for both binary alloy and random site energy disorder models. For the realistic case, which is best described by a binary alloy model without particle-hole symmetry, we predict density-of-states suppression below an energy scale which appears to be correlated with the corresponding single-impurity resonance., Comment: 5 pages, 4 figures