At low-energy, the band structure of graphene can be approximated by two degenerate valleys $(K,K^{\prime})$ about which the electronic spectra of the valence and conduction bands have linear dispersion relations. An electronic state in this band spectrum is a linear superposition of states from the $A$ and $B$ sublattices of the honeycomb lattice of graphene. In a quantizing magnetic field, the band spectrum is split into Landau levels with level N=0 having zero weight on the $B(A)$ sublattice for the $% K(K^{\prime})$ valley. Treating the valley index as a pseudospin and assuming the real spins to be fully polarized, we compute the energy of Wigner and Skyrme crystals in the Hartree-Fock approximation. We show that Skyrme crystals have lower energy than Wigner crystals \textit{i.e.} crystals with no pseudospin texture in some range of filling factor $ν$ around integer fillings. The collective mode spectrum of the valley-skyrmion crystal has three linearly-dispersing Goldstone modes in addition to the usual phonon mode while a Wigner crystal has only one extra Goldstone mode with a quadratic dispersion. We comment on how these modes should be affected by disorder and how, in principle, a microwave absorption experiment could distinguish between Wigner and Skyrme crystals.
The ground state of the two-dimensional electron systems in Bernal bilayer and ABC-stacked multilayer graphenes in the presence of a strong magnetic field is investigated with the Hartree-Fock approximation. Phase diagrams of the systems are obtained, focusing on charge density wave states including states with vortices of valley pseudospins (called a Skyrme crystal). The single-electron states in these stacked graphenes are given by two-component wave functions. That of the first excited Landau level has the same component as the lowest Landau level of the ordinary two-dimensional electrons. …
It is generally believed that a Wigner Crystal in single layer graphene can not form because the magnitudes of the Coulomb interaction and the kinetic energy scale similarly with decreasing electron density. However, this scaling argument does not hold for the low energy states in bilayer graphene. We consider the formation of a Wigner Crystal in weakly doped bilayer graphene with an energy gap opened by a perpendicular electric field. We argue that in this system the formation of the Wigner Crystal is not only possible, but different phases of the crystal with very peculiar properties may exi…
Excitation of electron-hole pairs in the vicinity of the Dirac cone by the Coulomb interaction gives rise to an asymmetric Breit-Wigner-Fano lineshape in the phonon Raman spectra in graphene. This asymmetric lineshape appears due to the interference effect between the phonon spectra and the electron-hole pair excitation spectra. The calculated Breit-Wigner-Fano asymmetric factor 1/qBWF as a function of the Fermi energy shows a V-shaped curve with a minimum value at the charge neutrality point and gives good agreement with the experimental result.
Using many-body configuration interaction techniques we show that Wigner crystallization occurs at the zigzag edges of graphene at surprisingly high electronic densities up to $0.8$ $\mbox{nm}^{-1}$. In contrast with one-dimensional electron gas, the flat-band structure of the edge states makes the system interaction dominated, facilitating the electronic localization. The resulting Wigner crystal manifests itself in pair-correlation functions, and evolves smoothly as the edge electron density is lowered. We also show that the crystallization affects the magnetization of the edges. While the e…
The single graphene layer is a novel material consisting of a flat monolayer of carbon atoms packed in a two-dimensional honeycomb-lattice, in which the electron dynamics is governed by the Dirac equation. A pseudo-spin phase-space approach based on the Wigner-Weyl formalism is used to describe the transport of electrons in graphene including quantum effects. Our full-quantum mechanical representation of the particles reveals itself to be particularly close to the classical description of the particle motion. We analyze the Klein tunneling and the correction to the total current in graphene in…
In spite of unscreened Coulomb interactions close to charge neutrality, relativistic massless electrons in graphene allegedly behave as noninteracting particles. A clue to this paradox is that both interaction and kinetic energies scale with particle density in the same way. In contrast, in a dilute gas of nonrelativistic electrons the different scaling drives the transition to Wigner crystal. Here we show that Dirac electrons in a graphene quantum dot with a mass gap localize à la Wigner for realistic values of device parameters. Our theoretical evidence relies on many-body observables obtain…
Graphene, with its massless linearly-dispersing carriers, in the quantum Hall regime provides an instructive comparison with conventional two-dimensional (2D) systems in which carriers have a nonzero band mass and quadratic dispersion. We investigate the influence of Landau level mixing in graphene on Wigner crystal states in the $n^\mathrm{th}$ Landau level obtained using single Landau level approximation. We show that the Landau level mixing does not qualitatively change the phase diagram as a function of partial filling factor $ν$ in the $n^\mathrm{th}$ level. We find that the inter-Landau …
Competition between liquid and solid states in two-dimensional electron system is an intriguing problem in condensed matter physics. We have investigated competing Wigner crystal and fractional quantum Hall ( FQH ) liquid phases in atomically thin suspended graphene devices in Corbino geometry. Low temperature magnetoconductance and transconductance measurements along with $IV$ characteristics all indicate strong charge density dependent modulation of electron transport. Our results show unconventional FQH phases which do not fit the standard Jain's series for conventional FQH states, instead …
Assuming that the nuclear magnetic resonance (NMR) signal from a $^{13}$C isotope enriched layer of graphene can be made sufficiently intense to be measured, we compute the NMR\ lineshape of the different crystals ground states that are expected to occur in graphene in a strong magnetic field. We first show that in nonuniform states, there is, in addition to the frequency shift due to the spin hyperfine interaction, a second contribution of equal importance from the coupling between the orbital motion of the electrons and the nuclei. We then show that, if the linewidth of the bare signal can b…
We study the possibility of Wigner crystallization in both single- and and bi-layer graphene using a real space tight binding model. In addition to verifying our earlier prediction for single layer graphene, we predict that the bilayer graphene can undergo a Wigner crystal transition at low enough carrier density thus offering a possibility of tunable and bipolar Wigner crystal. We find that aside from the Coulomb interaction of the carriers of two layers that stabilizes the charge ordered state, so does the inter layer coupling.
Graphene, a single free-standing sheet of graphite with honeycomb lattice structure, is a semimetal with carriers that have linear dispersion. A consequence of this dispersion is the absence of Wigner crystallization in graphene, since the kinetic and potential energies both scale identically with the density of carriers. We study the ground state of graphene in the presence of a strong magnetic field focusing on states with broken translational symmetry. Our mean-field calculations show that at integer fillings a uniform state is preferred, whereas at non-integer filling factors Wigner crysta…
We discuss the tight-binding models of solid state physics with the $Z_2$ sublattice symmetry in the presence of elastic deformations, and their important particular case -the tight binding model of graphene. In order to describe the dynamics of electronic quasiparticles we explore Wigner-Weyl formalism. It allows to calculate the two-point Green's function in the presence of both slowly varying external electromagnetic fields and the inhomogeneous modification of the hopping parameters resulted from the elastic deformations. The developed formalism allows us to consider the influence of elast…
We study the fractional quantum Hall effect in the central Landau level of bilayer graphene. By tuning the external applied magnetic field and the electric bias between the two layers one can access a regime where there is a degeneracy between Landau levels with orbital characters corresponding to N=0 and N=1 Galilean Landau levels. While the Laughlin state is generically the ground state for filling $ν=1/3$ we find that it can be destroyed and replaced by a Wigner crystal at the same filling factor by tuning the bias and applied field. This competition does not take place at $ν=2/3$ where the…
When a single molecule is connected to external electrodes by linker groups, the connectivity of the linkers to the molecular core can be controlled to atomic precision by appropriate chemical synthesis. Recently, the connectivity dependence of the electrical conductance and Seebeck coefficient of single molecules has been investigated both theoretically and experimentally. Here we study the connectivity dependence of the Wigner delay time of single-molecule junctions and the connectivity dependence of superconducting proximity effects, which occur when the external electrodes are replaced by …
We investigate the effects of coupling between the two zigzag edges of graphene nanoribbons on the Wigner crystallization of electrons and holes using a combination of tight-binding, mean field Hubbard and many-body configuration interaction methods. We show that the thickness of the nanoribbon plays a crucial role in the formation of Wigner crystal. For ribbon widths smaller than 16 \mboxÅ, increased kinetic energy overcomes the long-range Coulomb repulsion and suppresses the Wigner crystallization. For wider ribbons up to 38 \mboxÅ wide, strong Wigner localization is observed for even number…
We devise a model to explain why twisted bi-layer graphene (TBLG) exhibits insulating behavior when $ν=2,3$ charges occupy a unit moiré cell, a feature attributed to Mottness, but not for $ν=1$, clearly inconsistent with Mott insulation. We compute $r_s=E_U/E_K$, where $E_U$ and $E_K$ are the potential and kinetic energies, respectively, and show that (i) the Mott criterion lies at a density $10^4$ higher than in the experiments and (ii) a transition to a series of Wigner crystalline states exists as a function of $ν$. We find, for $ν=1$, $r_s$ fails to cross the threshold ($r_s = 37$) for the…
Moiré superlattices are emerging as a new route for engineering strongly correlated electronic states in two-dimensional van der Waals heterostructures, as recently demonstrated in the correlated insulating and superconducting states in magic-angle twisted bilayer graphene and ABC trilayer graphene/boron nitride moiré superlattices. Transition metal dichalcogenide (TMDC) moiré heterostructures provide another exciting model system to explore correlated quantum phenomena, with the addition of strong light-matter interactions and large spin-orbital coupling. Here we report the optical detection …
Divergent density of states offers the unique opportunity to explore a wide variety of correlated electron physics. In the thinnest limit, this has been predicted and verified in the ultra-flat bands of magic-angle twisted bilayer graphene, the band touching points of few-layer rhombohedral graphite, and the lightly doped rhombohedral trilayer graphene. The simpler and seemingly better understood Bernal bilayer graphene is also susceptible to orbital magnetism-driven phases at charge neutrality, such as layer antiferromagnet and quantum anomalous Hall octet. Here we report the discovery of a c…
Competing Orders Driven by Wigner Crystal Phase in Rhombohedral Graphene
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院物理研究所）；SPEC, CEA, CNRS, Université Paris-Saclay（法国原子能和替代能源委员会、国家科学研究中心、萨克雷大学SPEC实验室）；Department of Physics, Florida State University（佛罗里达州立大学物理系）；National High Magnetic Field Laboratory（国家高磁场实验室）；FSU Quantum Initiative, Florida State University（佛罗里达州立大学FSU量子倡议项目）；Research Center for Functional Materials, National Institute for Materials Science（日本物质材料研究机构功能材料研究中心）
We report a dual resonance feature in ballistic conductance through a quantum Hall graphene nanoribbon with a magnetic quantum dot. Such a magnetic quantum dot localizes Dirac fermions exhibiting anisotropic eigenenergy spectra with broken time-reversal symmetry. Interplay between the localized states and quantum Hall edge states is found to be two-fold, showing Breit-Wigner and Fano resonances, which is reminiscent of a double quantum dot system. By fitting the numerical results with the Fano-Breit-Wigner lineshape from the double quantum dot model, we demonstrate that the two-fold resonance …
We study $N$ interacting massless Dirac fermions confined in a two-dimensional quantum dot. Physical realizations of this problem include a graphene monolayer and the surface state of a strong topological insulator. We consider both a magnetic confinement and an infinite mass confinement. The ground state energy is computed as a function of the effective interaction parameter $α$ from the Hartree-Fock approximation and, alternatively, by employing the Müller exchange functional. For N=2, we compare those approximations to exact diagonalization results. The Hartree-Fock energies are highly accu…
Transport experiments on the two-dimensional electron gas (2DEG) confined into a semiconductor quantum well and subjected to a quantizing magnetic field have uncovered a rich variety of uniform and nonuniform phases such as the Laughlin liquids, the Wigner, bubble and Skyrme crystals and the quantum Hall stripe state. Optically pumped nuclear magnetic resonance (OP-NMR) has also been extremely useful in studying the magnetization and dynamics of electron solids with exotic spin textures such as the Skyrme crystal. Recently, it has been demonstrated that a related technique, resistively-detecte…
The interaction between electrons and plasmons in trilayer graphene is investigated within the Overhauser approach resulting in the 'plasmaron' quasi-particle. This interaction is cast into a field theoretical problem, nd its effect on the energy spectrum is calculated using improved Wigner-Brillouin perturbation theory. The plasmaron spectrum is shifted with respect to the bare electron spectrum by $ΔE(\mathbf{k})\sim 50\div200\,{\rm meV}$ for ABC stacked trilayer graphene and for ABA trilayer graphene by $ΔE(\mathbf{k})\sim 30\div150\,{\rm meV}$ ($ΔE(\mathbf{k})\sim 1\div5\,{\rm meV}$) for t…
We derive a quantum kinetic model describing the dynamics of graphene electrons in phase space based on the Wigner--Weyl formalism. To take into account the quantum nature of the carriers, we make use of the quantum Liouville equation for the density matrix. By relating the density matrix elements with the Wigner function, the equation of motion for the latter is established, with the Coulomb interaction being introduced self-consistently (i.e., in the Hartree approximation). The long-wavelength limit for the plasmon dispersion relation is obtained, for both ungated and gated situations. As an…
A graphene bilayer in a transverse magnetic field has a set of Landau levels with energies $E=\pm \sqrt{N(N+1)}\hslash ω_{c}^{\ast}$ where $ω_{c}^{\ast}$ is the effective cyclotron frequency and $% N=0,1,2,...$ All Landau levels but N=0 are four times degenerate counting spin and valley degrees of freedom. The Landau level N=0 has an extra degeneracy due to the fact that orbitals $n=0$ and $n=1$ both have zero kinetic energies. At integer filling factors, Coulomb interactions produce a set of broken-symmetry states with partial or full alignement in space of the valley and orbital pseudospins.…
The Wigner crystal state, first predicted by Eugene Wigner in 1934, has fascinated condensed matter physicists for nearly 90 years2-14. Studies of two-dimensional (2D) electron gases first revealed signatures of the Wigner crystal in electrical transport measurements at high magnetic fields2-4. More recently optical spectroscopy has provided evidence of generalized Wigner crystal states in transition metal dichalcogenide (TMDC) moiré superlattices. Direct observation of the 2D Wigner crystal lattice in real space, however, has remained an outstanding challenge. Scanning tunneling microscopy (S…
We consider the fate of the Wigner crystal state in a two dimensional system of massive Dirac electrons as the effective fine structure constant $α$ is increased. In a Dirac system, larger $α$ naively corresponds to stronger electron-electron interactions, but it also implies a stronger interband dielectric response that effectively renormalizes the electron charge. We calculate the critical density and critical temperature associated with quantum and thermal melting of the Wigner crystal state using two independent approaches. We show that at $α\gg 1$, the Wigner crystal state is best underst…
Recent experimental advances in the reconstruction of the Wigner function (WF) for electronic systems have led us to consider the possibility of employing this theoretical tool in the analysis of electron dynamics of uniaxially strained graphene. In this paper, we study the effect of strain on the WF of electrons in graphene under the interaction of a uniform magnetic field. This mechanical deformation modifies drastically the shape of the Wigner function of Landau and coherent states. The WF has a different behavior straining the material along the zigzag direction in comparison with the armc…
We uncover the interlayer shear mode of multi-layer graphene samples, ranging from bilayer-graphene (BLG) to bulk graphite, and show that the corresponding Raman peak measures the interlayer coupling. This peak scales from~43cm-1 in bulk graphite to~31cm-1 in BLG. Its low energy makes it a probe of near-Dirac point quasi-particles, with a Breit-Wigner-Fano lineshape due to resonance with electronic transitions. Similar shear modes are expected in all layered materials, providing a direct probe of interlayer interactions
We analyze interaction effects on boundary states of single layer graphene. Near a half filled band, both short and long-ranged interactions lead to a fully spin polarized configuration. In addition, the band of boundary states acquires a finite dispersion as function of the momentum parallel to the edge, induced by the interactions. Away from half filling the wavefunction develops charge correlations similar to those in a Wigner crystal, and the spin strongly alternates with the occupation of the boundary states. For certain fillings the ground state has a finite linear momentum, leading to t…
We propose a platform for flat Chern band by subjecting two-dimensional Dirac materials -- such as graphene and topological insulator thin films -- to a periodic magnetic field, which can be created by the vortex lattice of a type-II superconductor. As a generalization of the $n=0$ Landau level, the flat band of Dirac fermion under a nonuniform magnetic field remains at zero energy, exactly dispersionless and topologically protected, while its local density of states is spatially modulated due to the magnetic field variation. In the presence of short-range repulsion, we find fractional Chern i…
Transition metal dichalcogenide-based moire superlattices exhibit very strong electron-electron correlations, thus giving rise to strongly correlated quantum phenomena such as generalized Wigner crystal states. Theoretical studies predict that unusual quasiparticle excitations across the correlated gap between upper and lower Hubbard bands can arise due to long-range Coulomb interactions in generalized Wigner crystal states. Here we describe a new scanning single-electron charging (SSEC) spectroscopy technique with nanometer spatial resolution and single-electron charge resolution that enables…
We establish topological nature of Hall conductivity of graphene and other $Z_2$ crystals in 2D and 3D in the presence of inhomogeneous perturbations. To this end the lattice Weyl-Wigner formalism is employed. The non-uniform mechanical stress is considered, along with spatially varying magnetic field. The relation of the obtained topological invariant to level counting is clarified.
Feasibility study of continuous electronic Pomeranchuk cooling with a flavor-degenerate Wigner crystal
RWTH Aachen University（亚琛工业大学）；JARA Fundamentals of Future Information Technology（JARA未来信息技术基础）；nd Institute of Physics, RWTH Aachen University（亚琛工业大学第二物理研究所）；HFML-FELIX（荷兰强磁场实验室FELIX中心）；Radboud University（拉德堡德大学）；Max Planck Institute for the Structure and Dynamics of Matter（马普物质结构与动力学研究所）；Center for Free Electron Laser Science（自由电子激光科学中心）；University of Hamburg（汉堡大学）；I. Institute for Theoretical Physics, University of Hamburg（汉堡大学第一理论物理研究所）；Institute for Theoretical Physics and Astrophysics, University of Würzburg（维尔茨堡大学理论与天体物理研究所）；Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich（于利希研究中心彼得·格林贝格研究所（PGI-9））
Tuning interactions between Dirac states in graphene has attracted enormous interest because it can modify the electronic spectrum of the two-dimensional material, enhance electron correlations, and give rise to novel condensed-matter phases such as superconductors, Mott insulators, Wigner crystals and quantum anomalous Hall insulators. Previous works predominantly focus on the flat band dispersion of coupled Dirac states from different graphene layers. In this work, we propose a new route to realizing flat band physics in monolayer graphene under a periodic modulation from substrates. We take…
Chiral superconductors and competing states across a Lifshitz transition in rhombohedral pentalayer graphene
Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交大东京大学研究院）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；University of Tennessee（田纳西大学）；Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）；National Institute for Materials Science（物质材料研究机构）；Hefei National Laboratory（合肥国家实验室）；Wuhan University（武汉大学）；Southern University of Science and Technology（南方科技大学）
We present a theory of electronic properties of gated triangular graphene quantum dots with zigzag edges as a function of size and carrier density. We focus on electronic correlations, spin and geometrical effects using a combination of atomistic tight-binding, Hartree-Fock and configuration interaction methods (TB+HF+CI) including long range Coulomb interactions. The single particle energy spectrum of triangular dots with zigzag edges exhibits a degenerate shell at the Fermi level with a degeneracy N_{edge} proportional to the edge size. We determine the effect of the electron-electron intera…
We calculate, within a self-consistent Hartree-Fock approximation, the local density of states for different electron crystals in graphene subject to a strong magnetic field. We investigate both the Wigner crystal and bubble crystals with M_e electrons per lattice site. The total density of states consists of several pronounced peaks, the number of which in the negative energy range coincides with the number of electrons M_e per lattice site, as for the case of electron-solid phases in the conventional two-dimensional electron gas. Analyzing the local density of states at the peak energies, we…
