Table of contents

It is our pleasure to publish the Proceedings of the International Workshop on Dirac Electrons in Solids held in University of Tokyo, Japan, for January 14-15, 2015. The workshop was organized by the entitled project which lasted from April 2012 to March 2015 with 10 theorists. It has been supported by a Grand-in-Aid for Scientific Research (A) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan.

The subjects discussed in the workshop include bismuth, organic conductors, graphene, topological insulators, new materials including Ca3PbO, and new directions in theory (superconductivity, orbital susceptibility, etc). The number of participants was about 70 and the papers presented in the workshop include four invited talks, 16 oral presentations, and 23 poster presentations.

Dirac electron systems appear in various systems, such as graphene, quasi-two-dimensional organic conductors, bismuth, surface states in topological insulators, new materials like Ca3PbO. In these systems, characteristic transport properties caused by the linear dispersion of Dirac electrons and topological properties, have been extensively discussed. In addition to these, there are many interesting research fields such as Spin-Hall effect, orbital diamagnetism due to interband effects, Landau levels characteristic to Dirac dispersion, anomalous interlayer transport phenomena and magnetoresistance, the effects of spin-orbit interaction, and electron correlation. The workshop focused on recent developments of theory and experiment of Dirac electron systems in the above materials.

We note that all papers published in this volume of Journal of Physics: Conference Series were peer reviewed. Reviews were performed by expert referees with professional knowledge and high scientific standards in this field. Editors made efforts so that the papers may satisfy the criterion of a proceedings journal published by IOP Publishing.

We hope that all the participants of the workshop enjoyed discussions and that these proceedings of the workshop help to extend the international research activities into Dirac Electrons in Solids in the future.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

We study the importance of interband effects on the orbital susceptibility of three bands α-T₃ tight-binding models. The particularity of these models is that the coupling between the three energy bands (which is encoded in the wavefunctions properties) can be tuned (by a parameter α) without any modification of the energy spectrum. Using the gauge-invariant perturbative formalism that we have recently developped[1], we obtain a generic formula of the orbital susceptibility of α-T₃ tight-binding models. Considering then three characteristic examples that exhibit either Dirac, semi-Dirac or quadratic band touching, we show that by varying the parameter a and thus the wavefunctions interband couplings, it is possible to drive a transition from a diamagnetic to a paramagnetic peak of the orbital susceptibility at the band touching. In the presence of a gap separating the dispersive bands, we show that the susceptibility inside the gap exhibits a similar dia to paramagnetic transition.

Topological phases can not only be protected by internal symmetries (e.g., time-reversal symmetry), but also by crystalline symmetries, such as reflection or rotation symmetry. Recently a complete topological classification of reflection symmetric insulators, superconductors, nodal semimetals, and nodal superconductors has been established. In this article, after a brief review of the classification of reflection-symmetry-protected semimetals and nodal superconductors, we discuss an example of a three-dimensional topological Dirac semimetal, which exhibits time-reversal symmetry as well as reflection and rotation symmetries. We compute the surface state spectrum of this Dirac semimetal and identify the crystal lattice symmetries that lead to the protection of the surface states. We discuss the implications of our findings for the stability of the Fermi arc surface states of the Dirac material Na₃Bi. Our analysis suggests that the Fermi arc of Na₃Bi is gapped except at time-reversal invariant surface momenta, which is in agreement with recent photoemission measurements.

Effective g-factor for multiband system with a strong spin-orbit interaction is investigated. A simple, general, and gauge-invariant formula for the effective g-factor is obtained on the basis of k · p theory. By taking into account the symmetry for the T-point in Brillouinzone of bismuth, the formula of the effective g-factor is further simplified. The obtained results give a reasonable explanation for the longstanding puzzle about the large anisotropic g-factor of holes in bismuth.

We investigate the Meissner effect (ME) of Dirac electrons in a superconducting state on the basis of Kubo formula, and clarify that Meissner kernel becomes finite by use of the inter-band contribution. This mechanism of the ME for Dirac electron is completely different from that for the electron in the usual metals. We also derive the result of the electron gas by taking the non-relativistic limit of Dirac Hamiltonian, and clarify that the diamagnetic term of the Meissner kernel can be regarded as the inter-band contribution between electrons and positrons in terms of the Dirac model.

Motivated by the puzzling optical conductivity measurements in graphene, we speculate on the possible role of strong electronic correlations on the two-dimensional Dirac fermions. In this work we employ the slave-particle method to study the excitations of the Hubbard model on honeycomb lattice, away from half-filling. Since the ratio U/t ≈ 3.3 in graphene is not infinite, double occupancy is not entirely prohibited and hence a finite density of doublonscan be generated. We therefore extend the Ioff-Larkin composition rule to include a finite density of doublons. We then investigate the role played by each of these auxiliary particles in the optical absorption of strongly correlated Dirac fermions.

Phosphorene, monolayer black phosphorus, is a highly anistropicmaterial, where the band structure is Dirac like in one direction and Schrödinger like in the other direction. We present several analytic formulas to demonstrate the electronic properties. The conductance is highly anisotropic reflecting the anisotropy of the band structure. We also deteminethe cyclotron motion in phosphoreneand Landau level quantization by using the Bohr-Sommerfeldquantization. We show that the Landau-level energy behaves as B^2/3 as a function of external magnetic field, which is highly contrasted to the case of graphene where the Landau-level energy behaves as √B.

We consider electron systems characterized by chirality, such as Dirac fermion systems and iron-based superconductors. We investigate the chirality effect on superconducting states in such a system. We show that chirality effect leads to a nodal structure in the superconducting gap function. The node creation mechanism depends on the wave vector q of the pairing interaction and vorticity that characterizes chirality of electrons. The node creation effect due to chirality is significant for the case of Dirac fermions with q = (π, 0) and for the case of iron-based superconductors with q = (π,π).

The bulk and surface electronic structures of a candidate three-dimensional Dirac material Ca₃PbO and its family are discussed especially focusing on the spin texture on the surface states. We first explain the basic features of the bulk band structure of Ca₃PbO, such as emergence of Dirac fermions near the Fermi energy, and compare it with the other known three-dimensional Dirac semimetals. Then, the surface bands and spin-texture on them are investigated in detail. It is shown that the surface bands exhibit strong momentum-spin locking, which may be useful in some application for spin manipulation, induced by a combination of the inversion symmetry breaking at the surface and the strong spin-orbit coupling of Pb atoms. The surface band structure and the spin-textures are sensitive to the surface types.

We discuss a conductivity of a three-dimensional gas of Dirac electrons in the direction of magnetic field in high magnetic fields. Analytical expressions for the zero temperature conductivity are calculated in the linear response theory using the basis of relativistic Landau levels. The impurity scattering is treated in the self-consistent Born approximation which gives Landau level broadening increasing linearly in magnetic field. We demonstrate that in the special case of zero-gap Dirac semimetal this leads to the magnetic field and temperature independent conductivity in high magnetic field limit.

Dirac electrons with a zero-gap state (ZGS) in organic conductor α-(BEDT- TTF)₂I₃ result from a fine tuning of the seven nearest neighbors transfer integrals (a₁,a₂,a₃,b₁,b₂,b₃,b₄) between the four molecules of the unit cell. In this work we show that for given moduli |a₁1,...|b₄1, the possibility of having Dirac electron with a ZGS at 3/4 (or 1/4) filling strongly depends on the specific configurations of signs of the seven transfer integral. More precisely it is possible to classify the sign configurations into essentially four classes determined by χ_a = sign(a₂a₃) and χ_b = sign(b₁b₂b₃b₄). Using extended numerics, we show that for both weak and large inhomogeneity in the moduli, the class (χ_a,χ_b) = (—, —) is the most favorable to find Dirac electrons with ZGS at 3/4 (or 1/4) filling. For the class (χ_a,χ_b) = (+, +) in the opposite case, we never found any ZGS at either 1/4 or 3/4 filling. The last two classes given by (χ_a,χ_b) = (+, —) and (χ_a,χ_b) = (—, +) corresponding to an intermediate situation; they allow for ZGS at 3/4 (resp. 1/4) filling but are much less favorable than class (χ_a, χ_b) = (—, —). As a matter of fact, all previous numerical studies of Dirac electrons and ZGS in α-(BEDT-TTF)₂I₃ correspond to class (χ_a,χ_b) = (—, +).

We examine the reflectance R of Dirac electrons of organic conductor α-(BEDT- TTF)₂I₃ with a tilted cone which is characterized by a degree of the tilting η(= 0.8) and the angle a measured from the applied electric field. The reflectance is calculated using the complex dielectric constant, ε₁ + iε₂, which is evaluated from the in-plane dynamical conductivity with a constant damping Γ. The bulk reflectance on the two-dimensional plane is estimated by taking account of an interlayer distance of organic conductor α-(BEDT-TTF)₂I₃. We compare a difference of R between α = 0(i) and π/2 (ii), which correspond to the tilting being parallel and perpendicular to the electric field, respectively. The frequency ω dependence of R is examined for several magnitudes of the chemical potential μ to find a crossover from the zero doping to a finite doping. With increasing ω, R decreases from unity. For the small doping with μ < Γ, a difference of R between (i) and (ii) is small, while, for the large doping Γ < μ, a noticeable difference of R between (i) and (ii) exists in a certain region of ω around 2μ. The behavior that R of (ii) is larger than that of (i) is enhanced by the doping. We discuss the case of the clean limit and the role of both the intraband excitation and the interband excitation in the presence of Γ.

We investigate the anisotropic spin motive force in α-(BEDT-TTF)₂I₃, which is a multi-layered massless Dirac fermion system under pressure. Assuming the interlayer antiferromagnetic interaction and the interlayer anisotropic ferromagnetic interaction, we numerically examine the spin ordered state of the ground state using the steepest descent method. The anisotropic interaction leads to the anisotropic spin ordered state. We calculate the spin motive force produced by the anisotropic spin texture. The result quantitatively agrees with the experiment.

We discuss a role of the localized π orbital, which exists around the defect, on the defect-induced Kondo effect in graphene by a numerical renormalization group study. We find that the localized π orbital assists this Kondo effect, and the Kondo temperature is sensitive to the broadening of the localized π orbital. Secondly, we focus on the negative magnetoresistance of this Kondo effect. In the experimental result, it has been shown that the negative magnetoresistance is ten times larger than the usual Kondo effect. In order to clarify the mechanism of the ''magnetic sensitive" Kondo effect, as a first step, we study an orbital magnetic field dependence of the localized n orbital by a tight-binding model with a Peierls phase. We find that as the magnetic field increases, the spectral width of the localized π orbital increases and the local DOS at the Fermi level decreases. Since the Kondo temperature is strongly dependent of the broadening of the localized π orbital, it is expected that this Kondo effect is sensitive to the orbital magnetic field as observed in the experiment.

In this study, we investigate the Kondo effect induced by the s-d interaction where the conduction bands are occupied by Dirac fermions. The Dirac fermion has the linear dispersion and is described typically by the Hamiltonian such as H_k = νk·σ + mσ₀ for the wave number k where σ_j are Pauli matrices and σ₀ is the unit matrix. We derived the formula of the Kondo temperature T_K by means of the Green's function theory for Green's functions including Dirac fermions and the localized spin. The T_K was determined from a singularity of Green's functions in the form T_K ∝ exp(—const/ρ|J|). The Kondo effect will disappear when the Fermi surface is point like.

Elemental bismuth and its compounds host strong spin-orbit interaction which is at the heart of topologically non-trivial alloys based on bismuth. These class of materials are described in terms of 4x4 matrices at each v point where spin and orbital labels of the underlying electrons are mixed. In this work we investigate the single impurity Anderson model (SIAM) within a mean field approximation to address the nature of local magnetic moment formation in a generic Dirac Hamiltonian. Despite the spin-mixing in the Hamiltonian, within the Hartree approximation it turns out that the impuritys Green function is diagonal in spin label. In the three dimensional Dirac materials defined over a bandwidth D and spin-orbit parameter γ, that hybridizes with impurity through V, a natural dimensionless parameter V²D/2πγ³ emerges. So neither the hybridization strength, V, nor the spin-orbit coupling γ, but a combination thereof governs the phase diagram. By tuning chemical potential and the impurity level, we present phase diagram for various values of Hubbard U. Numerical results suggest that strong spin-orbit coupling enhances the local moment formation both in terms of its strength and the area of the local moment region. In the case that we tune the chemical potential in a similar way as normal metal we find that magnetic region is confined to μ ≥ ε₀, in sharp contrast to 2D Dirac fermions. If one fixes the chemical potential and tunes the impurity level, phase diagram has two magnetic regions which corresponds to hybridization of impurity level with lower and upper bands.

We theoretically study the quasi bound state of Dirac electrons in cylindrically symmetric quantum dots with sharp boundary. According to the existing picture, due to Klein tunneling "relativistic electrons" can not be localized by any confinement potential. We show however that despite of Klein tunneling, interference effects can cause the trapping of electron in quantum dots. Considering the quasi bound state as the state with complex energy, to find the energy of this state we solve the wave-equation with outgoing boundary condition at infinity. The imaginary part of complex energy determines the trapping time of electron within the quantum dot. We show that for any finite confining potential corresponding to any set of quantum numbers (n, m) where n is the principal quantum number and m the magnetic quantum number, there exists a continuous band of states with finite life time. Upper and lower edges of each band corresponds to infinitely long lived states trapped inside and outside the wall of the same radius. We term this phenomenon the intrinsic broadening as it is not caused by scattering from any external potential, nor by many-body effects. This broadening appears to arise from a combination of relativistic and interference effects. The imaginary part of energy which is different for energies along the energy band is controlled by the orbital angular momentum of electron and the depth of the confining potential.

We theoretically study the stability of a Weyl semimetal against strong long-range Coulomb interactions. We consider a lattice model for a time-reversal symmetry broken Weylsemimetal with two nodes, and take into account the 1/r Coulomb interactions between the bulk electrons. Based on the U(1) lattice gauge theory, we investigate the system from the strong coupling limit. It is shown that spatial inversion (parity) symmetry of the system is spontaneously broken in the strong coupling limit, and a Weylsemimetal with broken time- reversal and inversion symmetries, which is different from the noninteractingone, survives in the strong coupling limit.

We investigate spin Hall effect in a three-dimensional Dirac electron system. We derive diffusion equations in the Keldysh formalism and define a spin current from a spin diffusion equation. We solve the obtained diffusion equations in a steady state in the bulk and find that the bulk spin Hall conductivity is non-zero, though that for the conventional spin current is zero.

We discuss the relationship between the quantum Hall conductance and a fractal energy band structure, Hofstadter butterfly, on a square lattice under a magnetic field. At first, we calculate the Hall conductance of Hofstadter butterfly on the basis of the linear responce theory. By classifying the bands into some groups with a help of continued fraction expansion, we find that the conductance at the band gaps between the groups accord with the denominators of fractions obtained by aborting the expansion halfway. The broadening of Landau levels is given as an account of this correspondance.

We study an anomalous Hall effect in the massive Dirac electron system with broken time reversal symmetry. Using the model Hamiltonian with the spin-orbit interaction and a split term which breaks time reversal symmetry, we calculate the energy band, the Berry curvature and the intrinsic Hall conductivity in an analytical way. We show that the nonzero Berry curvature appears and thus an intrinsic Hall conductivity occurs. This anomalous Hall effect can be observed in such systems as a ferromagnetic Dirac electron system or a ferromagnet-coated Dirac electron system.

Symmetric graphene tunnelingfield-effect transistors (SymFETs) consisting of two independently gated graphene layers separated by a potential barrier have been proposed as a room-temperature resonant tunneling device. In a SymFET, the inelastic-coherent length (L_ϕ) of the electrons is considered to be one of the important characteristic scattering lengths. Weak localization (WL) is a good tool to estimate L_ϕ In this study, the surface of the chemical-vapor- deposited single-layer graphene was Ticleaned after performing the conventional fabrication processes for graphene transistors. We found that the charge-neutral point (V_CNP) shifted to a lower back-gate voltage and the mobility increased owing to Ticleaning. Ti-cleaned Hall bars were investigated at 0.3 K under magnetic fields of up to 14 T. Negative magnetoresistance(MR) appears because of the WL effect, and the MR increases as the back-gate voltage (V_G) approaches V_CNP. From a fitting analysis using the theoretical formulation of WL, we found that L_ϕ was greater than 100 nm and that L_ϕ decreased as V_G approached V_CNP because of electron-hole puddles and electron-electron interaction.

The effective g-factor and the selection rules for the matrix elements of the velocity operator are investigated for the Luttinger Hamiltonian under a magnetic field. It is shown that the g-factor has a strong k_z-dependence, where k_z is the wave vector along the magnetic field. This anomalous k_z-dependence is due to the interband effect of the magnetic field between the light and heavy hole bands. It is found that a nontrivial contribution to the longitudinal magnetoconductivity arises also due to the interband effects by analyzing the selection rules. This nontrivial interband contribution can be a source of the longitudinal spin current.

The magnetoconductivity of bismuth is theoretically investigated on the basis of the Kubo formula to interpret the longstanding mystery of linear magnetoresistance in bismuth. First, the magnetoconductivity for the isotropic Dirac model is studied. It is found that the inverse magnetoconductivity increases quadratically at low magnetic fields and is saturated at high fields. This high field property is in contrast to that obtained by the semiclassical theory, where the inverse magnetoconductivity keeps quadratic increase. Next, the magnetoconductivity for the extended Dirac model, which is a realistic model for bismuth, is studied. The inverse magnetoconductivity so obtained is not saturated, but is reduced at high fields. Implications of present results to the linear magnetoresistivity of bismuth are discussed.

By using the variationalcluster approximation and cluster perturbation theory, we investigate the magnetism and single-particle excitations of a periodic Anderson model on the honeycomb lattice as an effective model for the single-side hydrogenated graphene, namely, graphone. We calculate the magnetic moment as a function of U (Coulomb interaction on impurity sites) with showing that the ground state is ferrimagneticfor any U > 0. We then calculate the single-particle excitations and show that the single-particle excitations are gapless and exhibit quadratic dispersion relation near the Fermi energy.

The band structures of PbTe and SnTeare theoretically investigated. First- principles calculation is carried out on the basis of the density functional theory within the generalized gradient approximations. Then, in order to study the fine band structures in the low energy region, the tight-binding model is introduced based on the most localized Wannierfunctions derived from the first-principles calculations. It is found that the topology of the equal energy surface is different between PbTe and SnTe due to the spin-orbit interaction. The information of the topologically different Fermi surface will be important to study the superconducting mechanism on doped PbTeand SnTe.