Dirac Fermion System Research Articles

Fermion zero-energy modes and fractional fermion numbers in a fractional vortex-fermion model

Topological materials have attracted much attention from both physicists and mathematicians recently. Topological properties are closely related to the fermion number (index) of Dirac fermions. The fermion number is given by the η invariant introduced by Atiyah, Padoti and Singer. We discuss a system of Dirac fermions interacting with a vortex and a kink. This system will be realized as a layered material of superconductors and topological insulators, where the Dirac fermion exists on the surface of the topological insulator. The fermion number is fractionalized and the fermion zero-energy excitation mode emerges when Dirac fermions interact with vortices and kinks. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable single-valued fermion zero-energy mode does not exist in the core of the half-flux quantum vortex.

Interlayer quantum transport in Dirac semimetal BaGa2

The quantum limit is quite easy to achieve once the band crossing exists exactly at the Fermi level (EF) in topological semimetals. In multilayered Dirac fermion systems, the density of Dirac fermions on the zeroth Landau levels (LLs) increases in proportion to the magnetic field, resulting in intriguing angle- and field-dependent interlayer tunneling conductivity near the quantum limit. BaGa2 is an example of a multilayered Dirac semimetal with its quasi-2D Dirac cone located at EF, providing a good platform to study its interlayer transport properties. In this paper, we report the negative interlayer magnetoresistance induced by the tunneling of Dirac fermions between the zeroth LLs of neighboring Ga layers in BaGa2. When the field deviates from the c-axis, the interlayer resistivity ρzz(θ) increases and finally results in a peak with the applied field perpendicular to the c-axis. These unusual interlayer transport properties are observed together in the Dirac semimetal under ambient pressure and are well explained by the model of tunneling between Dirac fermions in the quantum limit.

Emergent symmetries and coexisting orders in Dirac fermion systems

The quantum phase diagram and critical behavior of two-dimensional Dirac fermions coupled to two compatible order-parameter fields with $O(N_1)\oplus O(N_2)$ symmetry is investigated. Recent numerical studies of such systems have reported evidence for non-Landau-Ginzburg-Wilson transitions and emergent $O(N_1+N_2)$ symmetry between the two ordered states, which has been interpreted within a scenario of deconfined quantum criticality in (2+1)-dimensional Dirac materials. Here, we provide two theoretical approaches to refine the phase diagrams of such systems. In the immediate vicinity of the multicritical point between the ordered phases and the semimetallic phase, we employ a non-perturbative field-theoretical analysis based on the functional renormalization group. For the particular case of $N_1=3$, $N_2=1$, we perform a large-scale quantum Monte Carlo analysis of the strong-coupling region, where both orders meet. Our findings support the robust emergence of enhanced symmetry at the multicritical point and suggest the transition between the two ordered phases to take place via a sequence of continuous transitions. In particular, we find that intermediate regimes of coexistence are present in the phase diagram for all values of $N_1$ and $N_2$.

Single-Carrier Transport in Graphene/hBN Superlattices.

Graphene/hexagonal boron nitride (hBN) moiré superlattices have attracted interest for use in the study of many-body effects and fractal physics in Dirac fermion systems. Many exotic transport properties have been intensively examined in such superlattices, but previous studies have not focused on single-carrier transport. The investigation of the single-carrier behavior in these superlattices would lead to an understanding of the transition of single-particle/correlated phenomena. Here, we show the single-carrier transport in a high-quality bilayer graphene/hBN superlattice-based quantum dot device. We demonstrate remarkable device controllability in the energy range near the charge neutrality point (CNP) and the hole-side satellite point. Under a perpendicular magnetic field, Coulomb oscillations disappear near the CNP, which could be a signature of the crossover between Coulomb blockade and quantum Hall regimes. Our results pave the way for exploring the relationship of single-electron transport and fractal quantum Hall effects with correlated phenomena in two-dimensional quantum materials.

Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System

Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings, and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is fractionalized and this is related to the presence of fermion zero-energy excitation modes. A zero-energy mode can be regarded as a Majorana fermion mode when the chemical potential vanishes. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable wave function of fermion zero-energy mode does not exist in the core of the half-flux quantum vortex. The index of Dirac operator and the fermion number have additional contributions when a soliton scalar field has a singularity.

Light-induced anomalous Hall effect in massless Dirac fermion systems and topological insulators with dissipation

Employing the quantum Liouville equation with phenomenological dissipation, we investigate the transport properties of massless and massive Dirac fermion systems that mimics graphene and topological insulators, respectively. The massless Dirac fermion system does not show an intrinsic Hall effect, but it shows a Hall current under the presence of circularly-polarized laser fields as a nature of a optically-driven nonequilibrium state. Based on the microscopic analysis, we find that the light-induced Hall effect mainly originates from the imbalance of photocarrier distribution in momentum space although the emergent Floquet–Berry curvature also has a non-zero contribution. We further compute the Hall transport property of the massive Dirac fermion system with an intrinsic Hall effect in order to investigate the interplay of the intrinsic topological contribution and the extrinsic light-induced population contribution. As a result, we find that the contribution from the photocarrier population imbalance becomes significant in the strong field regime and it overcomes the intrinsic contribution. This finding clearly demonstrates that intrinsic transport properties of materials can be overwritten by external driving and may open a way to ultrafast optical-control of transport properties of materials.

Specific heat and pairing of Dirac composite fermions in the half-filled Landau level

A recent proposal argues that an alternate description of the half-filled Landau level is a theory of massless Dirac fermions. We examine the possibility of pairing of these Dirac fermions by numerically solving the coupled Eliashberg equations unlike our previous calculation (Wang and Chakravarty, 2016). In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian and anti-Pfaffian states. However, in our approximation scheme, pairing is not possible in the putative particle–hole symmetric state for ℓ=0 angular momentum. The specific heat at low temperatures of a system of massless Dirac fermions interacting with a transverse gauge field, expected to be relevant for the half-filled Landau level, is calculated. Using the Luttinger formula, it is found to be ∝TlnT in the leading low temperature limit, due to the exchange of transverse gauge bosons. The result agrees with the corresponding one in the nonrelativistic composite fermion theory of Halperin, Lee and Read of the half-filled Landau level.

The transport properties in graphene/single-unit-cell cuprates van der Waals heterostructure

Inducing superconductivity into graphene by a proximity effect offers a chance to study exotic physics phenomena involved in Cooper pairs on a Dirac fermions system. Here we fabricate the monolayer graphene/single-unit-cell Bi2Sr2CaCu2O8+x (single-unit-cell Bi2212) van der Waals heterostructure, where the atomically flat surface from mechanically exfoliated single-unit-cell Bi2212 is the key to the pronounced proximity effect. The transport measurements in combination with direct and alternating current are performed on graphene/single-unit-cell Bi2212. The superconductivity is observed at the graphene/single-unit-cell Bi2212 heterostructure below 86 K. In the superconducting state, the nonlinear drain current–voltage relation is seen reflecting the characterisation of the Dirac–fermionic band structure of graphene. While for a lower bias, the differential conductance spectra shows a small subgap up to 60 K, implying a proximity-induced high-temperature superconductivity in graphene. Our results open a new way to study the superconductivity in two-dimensional Dirac fermion systems.

Perfect Crossed Andreev Reflection in Dirac Hybrid Junctions in the Quantum Hall Regime

Perfect crossed Andreev reflection (CAR) is striking for high-efficiency Cooper pair splitting, which bears promising applications in quantum communication. Recent experimental advances have disclosed the way to explore CAR in Dirac fermion systems under ultrastrong magnetic fields. We develop a scattering approach to study quantum-Hall-superconductor-quantum-Hall junctions formed by a two-dimensional time-reversal symmetric Dirac semimetal. We propose two different setups of the hybrid junction in the quantum limit, where only zeroth Landau levels are involved in transport to exploit perfect CAR. In both setups, the CAR probability can reach unity without applying bias voltage and is controllable by the magnetic field strength, the junction width, the length, and the doping of the superconductor. CAR dominates the nonlocal transport and is directly measurable by the differential conductances. We also identify a quantized spin injection per CAR event in one of the two setups. Our proposal is experimentally feasible and will be helpful for exploring high-efficiency Cooper pair splitting and spin injection in Dirac materials.

Nodal arc of disordered Dirac fermions and non-Hermitian band theory

We show that Dirac fermion systems in two dimensions generally exhibit disorder-induced nodal arc replacing the nodal point and tilted Dirac cone, provided that the two components of the Dirac fermion correspond to two distinct orbitals unrelated by symmetry. This result is explicitly demonstrated using renormalization group analysis in a disordered Dirac model that we introduce, where the disorder potential acts differently on the two orbitals. As we show by numerical simulations and self-consistent Born approximation calculation, this drives the system into a new strongly disordered phase.

Robust scheme for magnetotransport analysis in topological insulators

The recent excitement about Dirac and Weyl fermion systems has renewed interest in magnetotransport properties of multi-carrier systems. However, the complexity of their analysis, even in the simplest two-carrier case, has hampered a good understanding of the underlying phenomena. Here we propose a new analysis scheme for two independent conduction channels, that strongly reduces previous ambiguities and allows to draw robust conclusions. This is demonstrated explicitly for the example of three-dimensional topological insulators. Their temperature and gate voltage-dependent Hall coefficient and transverse magnetoresistance behavior, including the phenomenon of huge linear transverse magnetoresistance, can be traced back to two conduction channels, with fully determined carrier concentrations and mobilities. We further derive an upper limit for the transverse magnetoresistance. Its violation implies field dependences in the electronic band structure or scattering processes, or the presence of more than two effective carrier types. Remarkably, none of the three-dimensional topological insulators or semimetals with particularly large transverse magnetoresistance violates this limit.

Shubnikov–de Haas oscillations in the three-dimensional Dirac fermion system Ca3PbO

Cubic antiperovskite ${\mathrm{Ca}}_{3}\mathrm{PbO}$ is a candidate for three-dimensional (3D) Dirac fermion systems that have recently emerged as a class of topological materials exhibiting linear energy dispersion in the bulk. We report magnetotransport and tunnel diode oscillation (TDO) measurements on Bi-doped ${\mathrm{Ca}}_{3}\mathrm{PbO}$ in magnetic fields up to 55 T and temperatures between 2 and 78 K. By observing the Shubnikov--de Haas (SdH) oscillations, we resolve the bulk 3D Fermi surface (FS) with distinctive features of Dirac fermions including linear magnetoresistance, light effective mass, and a nontrivial phase shift. TDO measurements under high fields reveal the existence of two primary SdH frequencies, one being twice the other. Together with the low effective mass, the oscillations of these two frequencies account for the emergence of the Landau level splitting that persists up to 43 K. The field-angular-dependence of the oscillation frequencies with three branches confirms that the FS is composed of three pairs of hole pockets with uniaxial anisotropy on the \ensuremath{\Gamma}-X path, as predicted for bulk ${\mathrm{Ca}}_{3}\mathrm{PbO}$ crystal by density functional theory calculation.

Photon-mediated electronic correlation effects in irradiated two-dimensional Dirac systems

Periodically driven systems can host many interesting phenomena. Two-dimensional Dirac systems irradiated by circularly polarized light are especially attractive thanks to the special absorption and emission of photons near Dirac cones. Here, letting the light travel in the two-dimensional plane, we treat the light-driven Dirac systems by using a unitary transformation, instead of usual Floquet theory, to capture the photon-mediated electronic correlation effects. In this approach, the direct electron–photon interaction terms can be removed and the resulting effective electron–electron interactions can produce important effects. The effective interactions can produce topological band structure in the case of irradiated 2D Dirac fermion system, and can lift the energy degeneracy of the Dirac cones for irradiated graphene. This method can be applied to other light-driven Dirac systems to investigate their photon-mediated electronic effects. These phenomena would be observed with ultraviolet light in some effective two-dimensional Dirac systems of honeycomb long-period superstructures.

Charge-Density-Wave Transitions of Dirac Fermions Coupled to Phonons.

The spontaneous generation of charge-density-wave order in a Dirac fermion system via the natural mechanism of electron-phonon coupling is studied in the framework of the Holstein model on the honeycomb lattice. Using two independent and unbiased quantum MonteCarlo methods, the phase diagram as a function of temperature and coupling strength is determined. It features a quantum critical point as well as a line of thermal critical points. Finite-size scaling appears consistent with fermionic Gross-Neveu-Ising universality for the quantum phase transition and bosonic Ising universality for the thermal phase transition. The critical temperature has a maximum at intermediate couplings. Our findings motivate experimental efforts to identify or engineer Dirac systems with sufficiently strong and tunable electron-phonon coupling.

Energy quantization at the three-quarter Dirac point in a magnetic field

The quantization of the energy in a magnetic field (Landau quantization) at a three-quarter Dirac point is studied theoretically. The three-quarter Dirac point is realized in the system of massless Dirac fermions with the critically tilted Dirac cone in one direction, where a linear term disappears and a quadratic term $\alpha_2 q_x^2$ with aconstant $\alpha_2$ plays an important role. The energy is obtained as $E_n \propto \alpha_2^{\frac{3}{5}} (n B)^{\frac{4}{5}}$, where $n=1, 2, 3, \dots$, by means of numerically and analytically solving the differential equation, as well as by the semiclassical quantization rule. The existence of the $n=0$ state is studied by introducing the energy gap due to the inversion-symmetry-breaking term, and it is obtained that the $n=0$ state exists in one of a pair of three-quarter Dirac points, depending on the direction of the magnetic field when the energy gap is finite.

Anisotropy of Dirac cones and Van Hove singularity in an organic Dirac fermion system

We propose an experimental method to examine the in-plane anisotropy of electronic structure in layered conductors. In the method, we measure the interlayer magnetoresistance as a function of in-plane magnetic field orientation. We applied it to an organic Dirac fermion system a-(BEDT-TTF)2I3 to experimentally determine the orientation of the anisotropic Dirac cones. It is concluded that the long axis of the elliptic constant-energy contours of the Dirac cone is tilted by approximately -30 deg from the crystalline a-axis to b-axis under hydrostatic pressures. Additionally, we observed a signature of van Hove singularity (which is a saddle point of the band dispersion) at 30-40 K above or below the Dirac point. The ridgeline of the saddle point is estimated as almost parallel to the crystalline b-axis.

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.

Design of 2D massless Dirac fermion systems and quantum spin Hall insulators based on sp\u2013sp2 carbon sheets

Grapheneis a massless Dirac fermion system, featuring Dirac points in momentum space. It was also first identified as a quantum spin Hall (QSH) insulator when considering spin–orbit coupling (SOC), which opens a band gap at the Dirac points. This discovery has initiated new research efforts to study the QSH effect, towards its application for quantum computing and spintronics. Although the QSH effect has been observed in HgTe quantum wells, the SOC strength of graphene is too small (~1 µeV) to induce the topological insulator phase in an experimentally achievable temperature regime. Here, we perform a systematic atomistic simulation to design two-dimensional sp–sp2 hybrid carbon sheets to discover new Dirac systems, hosting the QSH phase. 21 out of 31 newly discovered carbon sheets are identified as Dirac fermion systems without SOC, distinct from graphene in the number, shape, and position of the Dirac cones occurring in the Brillouin zone. Moreover, we find 19 out of the 21 new Dirac fermion systems become QSH insulators with a sizable SOC gap enhanced up to an order of meV, thus allowing for the QSH effect at experimentally accessible temperatures. In addition, based on the 26 Dirac fermion systems, we make a connection between the number of Dirac points without SOC and the resultant QSH phase in the presence of SOC. Our findings present new prospects for the design of topological materials with desired properties.

Condition for the emergence of a bulk Fermi arc in disordered Dirac-fermion systems

We present a renormalization group analysis of the disorder effects on the low-energy behaviors of two-dimensional tilted Dirac-fermion systems, in which the fermions have two distinct orbitals unrelated by any symmetry. Four types of disordered potential, two interorbital and two intraorbital, are considered. If there is only one type of interorbital disorder, the fermion-disorder scattering induces logarithmic or power-law corrections to the fermion density of states and specific heat. In contrast, the intraorbital disorder can turn the system into a strongly disordered phase. In this disordered phase, calculations based on self-consistent Born approximation reveal that the Dirac point is destroyed and replaced by a bulk Fermi arc. We also study the interplay of four types of disorder, and find that the Dirac point can either remain intact or give place to a Fermi arc. We obtain the condition for the emergence of a Fermi arc in this case. Our results indicate that disorders can result in rich low-energy properties of tilted Dirac fermions.

The role of electron-electron interactions in two-dimensional Dirac fermions.

The role of electron-electron interactions in two-dimensional Dirac fermion systems remains enigmatic. Using a combination of nonperturbative numerical and analytical techniques that incorporate both the contact and long-range parts of the Coulomb interaction, we identify the two previously discussed regimes: a Gross-Neveu transition to a strongly correlated Mott insulator and a semimetallic state with a logarithmically diverging Fermi velocity accurately described by the random phase approximation. We predict that experimental realizations of Dirac fermions span this crossover and that this determines whether the Fermi velocity is increased or decreased by interactions. We explain several long-standing mysteries, including why the observed Fermi velocity in graphene is consistently about 20% larger than values obtained from ab initio calculations and why graphene on different substrates shows different behaviors.