Two-dimensional Dirac Fermions Research Articles

Response to Comment on "The role of electron-electron interactions in two-dimensional Dirac fermions".

Hesselmann et al question one of our conclusions: the suppression of Fermi velocity at the Gross-Neveu critical point for the specific case of vanishing long-range interactions and at zero energy. The possibility they raise could occur in any finite-size extrapolation of numerical data. Although we cannot definitively rule out this possibility, we provide mathematical bounds on its likelihood.

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

Tang et al (Research Articles, 10 August 2018, p. 570) report on the properties of Dirac fermions with both on-site and Coulomb interactions. The substantial decrease, up to ~40%, of the Fermi velocity of Dirac fermions with on-site interaction is inconsistent with the numerical data near the Gross-Neveu quantum critical point. This results from an inappropriate finite-size extrapolation.

Anomalous Goos-Hänchen shift in the Floquet scattering of Dirac fermions

We study the inelastic scattering of two-dimensional massless Dirac fermions by an inhomogeneous time-dependent driving field. As a physical realization we consider a monolayer graphene normally illuminated with a circularly polarized laser of frequency $\Omega$ in a given region. The interaction Hamiltonian introduced by the laser, being periodic in time, can be treated with the Floquet method which naturally leads to a multi-channel scattering problem. We analyze planar and circular geometries of the interface separating the irradiated and non-irradiated regions and find that there is an anomalous Goos-H\"anchen shift in the inelastic channel. The latter is independent of the amplitude of the driving while its sign is determined by the polarization of the laser field. We related this shift with the appearance of topological edge states between two illuminated regions of opposite chiralities.

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.

Excitonic pairing of two-dimensional Dirac fermions near the antiferromagnetic quantum critical point

Two-dimensional Dirac fermions are subjected to two types of interactions, namely the long-range Coulomb interaction and the short-range on-site interaction. The former induces excitonic pairing if its strength $\alpha$ is larger than some critical value $\alpha_c$, whereas the latter drives an antiferromagnetic Mott transition when its strength $U$ exceeds a threshold $U_c$. Here, we study the impacts of the interplay of these two interactions on excitonic pairing with the Dyson-Schwinger equation approach. We find that the critical value $\alpha_c$ is increased by weak short-range interaction. As $U$ increases to approach $U_c$, the quantum fluctuation of antiferromagnetic order parameter becomes important and interacts with the Dirac fermions via the Yukawa coupling. After treating the Coulomb interaction and Yukawa coupling interaction on an equal footing, we show that $\alpha_c$ is substantially increased as $U \rightarrow U_c$. Thus, the excitonic pairing is strongly suppressed near the antiferromagnetic quantum critical point. We obtain a global phase diagram on the $U$-$\alpha$ plane, and illustrate that the excitonic insulating and antiferromagnetic phases are separated by an intermediate semimetal phase. These results provide a possible explanation of the discrepancy between recent theoretical progress on excitonic gap generation and existing experiments in suspended graphene.

Selective trapping of hexagonally warped topological surface states in a triangular quantum corral.

The surface of a three-dimensional topological insulator (TI) hosts two-dimensional massless Dirac fermions (DFs), the gapless and spin-helical nature of which leads to their high transmission through surface defects or potential barriers. Here, we report the behaviors of topological surface states (TSS) in a triangular quantum corral (TQC) which, unlike a circular corral, is supposed to be totally transparent for DFs. By real-space mapping of the electronic structure of TQCs, both the trapping and detrapping behaviors of the TSS are observed. The selection rules are found to be governed by the geometry and spin texture of the constant energy contour of TSS upon the strong hexagonal warping in Bi2Te3. Our work indicates the extended nature of TSS and elucidates the selection rules of the trapping of TSS in the presence of a complicated surface state structure, giving insights into the effective engineering of DFs in TIs.

Non-Hermitian quantum systems and their geometric phases

We discuss the basic theoretical framework for non-Hermitian quantum systems with particular emphasis on the diagonalizability of non-Hermitian Hamiltonians and their $GL(1,\mathbb{C})$ gauge freedom, which are relevant to the adiabatic evolution of non-Hermitian quantum systems. We find that the adiabatic evolution is possible only when the eigen-energies are real. The accompanying geometric phase is found to be generally complex and associated with not only the phase of a wavefunction but also its amplitude. The condition for the real geometric phase is laid out. Our results are illustrated with two examples of non-Hermitian $\mathcal{PT}$ symmetric systems, the two-dimensional non-Hermitian Dirac fermion model and bosonic Bogoliubov quasi-particles.

Fractionalization Waves in Two-Dimensional Dirac Fermions: Quantum Imprint from One Dimension.

Particle fractionalization is believed to orchestrate the physics of many strongly correlated systems, yet its direct experimental detection remains a challenge. We propose a simple measurement for an ultracold matter system, in which correlations in initially decoupled 1D chains are imprinted via quantum quench upon two-dimensional Dirac fermions. Luttinger liquid correlations launch relativistic "fractionalization waves" along the chains, while coupling noninteracting chains induces perpendicular dispersion. These could be easily distinguished in an ultracold gas experiment.

Microscopic theory of spin relaxation anisotropy in graphene with proximity-induced spin-orbit coupling

Inducing sizable spin--orbit interactions in graphene by proximity effect is establishing as a successful route to harnessing two-dimensional Dirac fermions for spintronics. Semiconducting transition metal dichalcogenides (TMDs) are an ideal complement to graphene because of their strong intrinsic spin--orbit coupling (SOC) and spin/valley-selective light absorption, which allows all-optical spin injection into graphene. In this study, we present a microscopic theory of spin dynamics in weakly disordered graphene samples subject to uniform proximity-induced SOC as realized in graphene/TMD bilayers. A time-dependent perturbative treatment is employed to derive spin Bloch equations governing the spin dynamics at high electronic density. Various scenarios are predicted, depending on a delicate competition between interface-induced Bychkov-Rashba and spin--valley (Zeeman-type) interactions and the ratio of intra- to inter-valley scattering rates. For weak SOC compared to the disorder-induced quasiparticle broadening, the anisotropy ratio of out-of-plane to in-plane spin lifetimes $\zeta=\tau_{s}^{\perp}/\tau_{s}^{\parallel}$ agrees qualitatively with a toy model of spins in a weak fluctuating SOC field recently proposed by Cummings and co-workers [PRL 119, 206601 (2017)]. In the opposite regime of well-resolved SOC, qualitatively different formulae are obtained, which can be tested in ultra-clean heterostructures characterized by uniform proximity-induced SOC in the graphene layer.

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.

Magnetic manifestation of discrete scaling symmetry in Dirac semimetals

Two-dimensional and three-dimensional massless Dirac fermions can form a sequence of quasibound states with an attractive charged impurity. These quasibound states exhibit a discrete scaling symmetry, i.e., the energy ratio between two successive states is a constant. Through the calculation of the energy spectrum directly, we find that in two dimension an applied magnetic field can shift or even destroy the quasibound states around the Dirac point and their discrete scaling symmetry disappears. However, as the magnetic field increases, the remaining quasibound states are pushed up to the Dirac point. When one quasibound state is close to the Dirac point, the spectrum is modified significantly, due to the resonant scattering. The magnetic oscillation of the spectrum displays the same discrete scaling symmetry as the quasibound state does in zero magnetic field. This phenomenon also occurs in the quantum limit of three-dimensional Dirac semimetals, where the system becomes quasi one-dimensional essentially. Our theoretical analysis are in good agreement with the recent experimental observations.

Evolution of electronic structure and electron-phonon coupling in ultrathin tetragonal CoSe films

Two-dimensional materials have attracted considerable research attention recently due to their extraordinary physical and chemical properties. In this paper, we systematically investigate the electronic structure of ultrathin tetragonal CoSe films with different thicknesses that are isostructural to two-dimensional high-temperature superconductor FeSe using high-resolution angle-resolved photoemission spectroscopy, scanning tunneling microscopy, and ab initio calculation. Interestingly, the nonsymmorphic symmetry of the CoSe monolayer protects both exotic saddle band degeneracy and Dirac points, realizing two-dimensional Dirac fermions. Our temperature and film-thickness-dependent measurements reveal momentum-dependent electron-phonon coupling in CoSe films, which is enhanced with reduced film thickness. Moreover, the electronic states in the CoSe monolayer strongly couple to high-frequency phonons in the $\mathrm{SrTi}{\mathrm{O}}_{3}$ substrate, similar to the situation in the FeSe monolayer. Our results not only present a platform to investigate the electronic properties in close vicinity of the high-temperature superconductor FeSe monolayer, but also shed light on the understanding of the coupling between two-dimensional materials and substrates beneath.

Two-dimensional Dirac fermion in presence of an asymmetric vector potential

We introduce the new, exactly solvable model of the two-dimensional Dirac fermion in presence of an asymmetric, Pöschl–Teller-like vector potential. Utilizing the translation invariance of the system, the effective one-dimensional stationary equation is brought into the form of the Heun equation and its fundamental solutions are found as an irreducible combination of two Gauss hypergeometric functions. The energy spectrum and the scattering is studied dependant on the conserved longitudinal momentum as well as on the strength of the coupling.

Collisionless Transport Close to a Fermionic Quantum Critical Point in Dirac Materials.

Quantum transport close to a critical point is a fundamental, but enigmatic problem due to fluctuations, persisting at all length scales. We report the scaling of optical conductivity (OC) in the collisionless regime (ℏω≫k_{B}T) in the vicinity of a relativistic quantum critical point, separating two-dimensional (d=2) massless Dirac fermions from a fully gapped insulator or superconductor. Close to such a critical point, gapless fermionic and bosonic excitations are strongly coupled, leading to a universal suppression of the interband OC as well as of the Drude peak (while maintaining its delta function profile) inside the critical regime, which we compute to the leading order in 1/N_{f}- and ε expansions, where N_{f} counts the fermion flavor number and ε=3-d. Correction to the OC at such a non-Gaussian critical point due to the long-range Coulomb interaction and generalizations of these scenarios to a strongly interacting three-dimensional Dirac or Weyl liquid are also presented, which can be tested numerically and possibly from nonperturbative gauge-gravity duality, for example.

Two-dimensional type-II Dirac fermions in a LaAlO3/LaNiO3/LaAlO3 quantum well

The type-II Dirac fermions that are characterized by a tilted Dirac cone and anisotropic magneto-transport properties have been recently proposed theoretically and confirmed experimentally. Here, we predict the emergence of two-dimensional type-II Dirac fermions in LaAlO3/LaNiO3/LaAlO3 quantum-well structures. Using first-principles calculations and model analysis, we show that the Dirac points are formed at the crossing between the dx2-y2 and dz2 bands protected by the mirror symmetry. The energy position of the Dirac points can be tuned to appear at the Fermi energy by changing the quantum-well width. For the quantum-well structure with a two-unit cell thick LaNiO3 layer, we predict the coexistence of the type-II Dirac points and the Dirac nodal line. The results are analyzed and interpreted using a tight-binding model and symmetry arguments. Our findings offer a practical way to realize the 2D type-II Dirac fermions in oxide heterostructures.

Two-dimensional type-II Dirac fermions in layered oxides

Relativistic massless Dirac fermions can be probed with high-energy physics experiments, but appear also as low-energy quasi-particle excitations in electronic band structures. In condensed matter systems, their massless nature can be protected by crystal symmetries. Classification of such symmetry-protected relativistic band degeneracies has been fruitful, although many of the predicted quasi-particles still await their experimental discovery. Here we reveal, using angle-resolved photoemission spectroscopy, the existence of two-dimensional type-II Dirac fermions in the high-temperature superconductor La1.77Sr0.23CuO4. The Dirac point, constituting the crossing of d_{x^2 - y^2} and d_{z^2} bands, is found approximately one electronvolt below the Fermi level (EF) and is protected by mirror symmetry. If spin-orbit coupling is considered, the Dirac point degeneracy is lifted and the bands acquire a topologically non-trivial character. In certain nickelate systems, band structure calculations suggest that the same type-II Dirac fermions can be realised near EF.

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.

Qualitative analysis of magnetic waveguides for two-dimensional Dirac fermions

We focus on the confinement of two-dimensional Dirac fermions within the waveguides created by realistic magnetic fields. Understanding of their band structure is of our main concern. We provide easily applicable criteria, mostly depending only on the asymptotic behavior of the magnetic field, that can guarantee existence or absence of the energy bands and provide valuable insight into the systems where analytical solution is impossible. The general results are employed in specific systems where the waveguide is created by the magnetic field of a set of electric wires or magnetized strips.

Sensitivity of the anomalous Hall effect to disorder correlations

Both longitudinal and anomalous Hall conductivity are computed in the model of two-dimensional Dirac fermions with a mass in the presence of arbitrary correlated weak disorder. The anomalous Hall conductivity is shown to be highly sensitive to the correlation properties of the random potential, such as the correlation length, while it remains independent of the integral disorder strength. This property extends beyond the Dirac model making the anomalous Hall effect an interesting tool to probe disorder correlations.

Two-Dimensional Dirac Fermions Protected by Space-Time Inversion Symmetry in Black Phosphorus.

We report the realization of novel symmetry-protected Dirac fermions in a surface-doped two-dimensional (2D) semiconductor, black phosphorus. The widely tunable band gap of black phosphorus by the surface Stark effect is employed to achieve a surprisingly large band inversion up to ∼0.6 eV. High-resolution angle-resolved photoemission spectra directly reveal the pair creation of Dirac points and their movement along the axis of the glide-mirror symmetry. Unlike graphene, the Dirac point of black phosphorus is stable, as protected by space-time inversion symmetry, even in the presence of spin-orbit coupling. Our results establish black phosphorus in the inverted regime as a simple model system of 2D symmetry-protected (topological) Dirac semimetals, offering an unprecedented opportunity for the discovery of 2D Weyl semimetals.