Unified Formula for Stationary Josephson Current in Planar Graphene Junctions

The stationary Josephson current in a ballistic graphene system is theoretically studied with focus on a planar junction consisting of a monolayer graphene sheet on top of which a pair of superconducting electrodes is deposited. To characterize such a planar junction, we employ two parameters: the coupling strength between the graphene sheet and the superconducting electrodes, and a potential drop induced in the graphene sheet by direct contact with the electrodes. We derive a general formula for the Josephson current by taking these parameters into account in addition to other basic parameters, such as temperature and chemical potential. The resulting formula applies to a wide range of parameters and reproduces previously reported results in certain limits.Comment: 7 pages, 4 figure

DMPK Equation for Transmission Eigenvalues in Metallic Carbon Nanotubes

The Dorokhov-Mello-Pereyra-Kumar (DMPK) equation for transmission eigenvalues is derived for metallic carbon nanotubes with several conducting channels when the potential range of scatterers is larger than the lattice constant. With increasing system length L, the system approaches a fixed point, where only one channel is perfectly conducting and other channels are completely closed. The asymptotic behavior of the conductance in the long-L regime is investigated on the basis of the DMPK equation. It is shown that the length scale for the exponential decay of the typical conductance is reduced due to the presence of the perfectly conducting channel. If a magnetic field is applied, the system falls into the unitary class. It is pointed out that this transition is characterized by the disappearance of the perfectly conducting channel and the increase in decay length for the typical conductance.Comment: 5 page

Quantum Electron Transport in Disordered Wires with Symplectic Symmetry

The conductance of disordered wires with symplectic symmetry is studied by the supersymmetric field theory. Special attention is focused on the case where the number of conducting channels is odd. Such a situation can be realized in metallic carbon nanotubes. The average dimensionless conductance g is obtained using Zirnbauer's super-Fourier analysis. It is shown that with increasing wire length, g approaches to 1 in the odd-channel case, while g vanishes in the ordinary even-channel case. It should be emphasized that the so-called Zirnbauer's zero mode, which has been believed to be unphysical, is essential for describing the anomalous behavior in the odd-channel case.Comment: 5 page

Regularization of a massless Dirac model to describe anomalous electromagnetic response of Weyl semimetals

An unbounded massless Dirac model with two nondegenerate Dirac cones is the simplest model for Weyl semimetals, which show the anomalous electromagnetic response of chiral magnetic effect (CME) and anomalous Hall effect (AHE). However, if this model is naively used to analyze the electromagnetic response within a linear response theory, it gives the result apparently inconsistent with the persuasive prediction based on a lattice model. We show that this serious difficulty is related to the breaking of current conservation in the Dirac model due to quantum anomaly and can be removed if current and charge operators are redefined to include the contribution from the anomaly. We demonstrate that the CME as well as the AHE can be properly described using newly defined operators, and clarify that the CME is determined by the competition between the contribution from the anomaly and that from low-energy electrons.Comment: 5 page

Chiral surface states on the step edge in a Weyl semimetal

A Weyl semimetal with a pair of Weyl nodes accommodates chiral states on its flat surface if the Weyl nodes are projected onto two different points in the corresponding surface Brillouin zone. These surface states are collectively referred to as a Fermi arc as they appear to connect the projected Weyl nodes. This statement assumes that translational symmetry is present on the surface and hence electron momentum is a conserved quantity. It is unclear how chiral surface states are modified if the translational symmetry is broken by a particular system structure. Here, focusing on a straight step edge of finite width, we numerically analyze how chiral surface states appear on it. It is shown that the chiral surface states are algebraically (i.e., weakly) localized near the step edge. It is also shown that the appearance of chiral surface states is approximately determined by a simple condition characterized by the number of unit atomic layers constituting the step edge together with the location of the Weyl nodes.Comment: 6 pages, 12 figure

Local Density of States in a Helical Tomonaga-Luttinger Liquid of Loop and Josephson Junction Geometries

The local density of states (LDOS) in a one-dimensional helical channel of finite length is studied within a Tomonaga-Luttinger model at zero temperature. Two particular cases of loop and Josephson junction geometries are considered. The LDOS, as a function of energy $\omega$ measured from the Fermi level, consists of equally spaced spikes of the $\delta$-function type, and electron-electron interactions modify their relative height. It is shown that, in the loop geometry, the height of spikes decreases as $\omega \to 0$ everywhere in the system. It is also shown that, in the Josephson junction, the behavior of the LDOS significantly depends on the spatial position. At the end points of the junction, the height increases as $\omega \to 0$ and its variation is more pronounced than that in the loop case. Away from the end points, the height of spikes shows a non-monotonic $\omega$-dependence, which disappears in the long-junction limit.Comment: 7 pages, 6 figure

Spontaneous charge current in a doped Weyl semimetal

A Weyl semimetal hosts low-energy chiral surface states, which appear to connect a pair of Weyl nodes in reciprocal space. As these chiral surface states propagate in a given direction, a spontaneous circulating current is expected to appear near the surface of a singly connected Weyl semimetal. This possibility is examined by using a simple model with particle-hole symmetry. It is shown that no spontaneous charge current appears when the Fermi level is located at the band center. However, once the Fermi level deviates from the band center, a spontaneous charge current appears to circulate around the surface of the system and its direction of flow is opposite for the cases of electron doping and hole doping. These features are qualitatively unchanged even in the absence of particle-hole symmetry. The circulating charge current is shown to be robust against weak disorder.Comment: 7 pages, 16 figure

Persistent current due to a screw dislocation in Weyl semimetals: Role of one-dimensional chiral states

A Weyl semimetal pierced by a screw dislocation accommodates one-dimensional (1D) chiral states along the corresponding dislocation line. As these states propagate in a particular direction determined by their chirality, a persistent current (i.e., charge current in equilibrium) is expected to appear in the interior of the system. To confirm this expectation, we numerically calculate the charge current in a Weyl semimetal in the presence of a screw dislocation. It is shown that a significant charge current is induced by the 1D chiral states near the dislocation. We also analyze the spatial distribution of the charge current focusing on the top and bottom surfaces of the system, at which the screw dislocation is terminated, and give an overview of how the charge current due to the dislocation is converted to that carried by other states near the termination point of the dislocation.Comment: 7 pages, 14 figure

Landau levels on a surface of weak topological insulators

A three-dimensional weak topological insulator (WTI), being equivalent to stacked layers of two-dimensional quantum spin-Hall insulators, accommodates massless Dirac electrons on its side surface. A notable feature of WTIs is that surface states typically consist of two Dirac cones in the reciprocal space. We study the Landau quantization of Dirac electrons of WTIs in a perpendicular magnetic field. It is shown that when the magnetic length $l_B$ is much larger than the interlayer distance $a$, surface electrons are quantized into Landau levels according to the ordinary quantization rule for Dirac electrons. It is also shown that, with decreasing $l_B$ toward $a$, each Landau level and its spin state become modulated in a nontrivial manner. We demonstrate that this is attributed to the mixing of two Dirac cones induced by the discreteness of the layered structure.Comment: 8 pages, 12 figure

Delocalization of surface Dirac electrons in disordered weak topological insulators

The spectrum of massless Dirac electrons on the side surface of a three-dimensional weak topological insulator is significantly affected by whether the number of unit atomic layers constituting the sample is even or odd; it has a finite-size energy gap in the even case while it is gapless in the odd case. The conductivity of such a two-dimensional Dirac electron system with quenched disorder is calculated when the Fermi level is located at the Dirac point. It is shown that the conductivity increases with increasing disorder and shows no clear even-odd difference when the aspect ratio of the system is appropriately fixed. From the system-size dependence of the average conductivity, the scaling function $\beta$ is determined under the one-parameter scaling hypothesis. The result implies that $\beta = 0$ in the clean limit at which the conductivity is minimized, and that $\beta > 0$ otherwise. Hence, the system is a perfect metal in the thermodynamic limit except in the clean limit that should be regarded as an unstable fixed point.Comment: 5 pages, 7 figure