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One of the intrinsic drift velocity limit of the quantum Hall effect is the collective magneto-exciton (ME) instability. It has been demonstrated in bilayer graphene (BLG) using noise measurements. We reproduce this experiment in monolayer graphene (MLG), and show that the same mechanism carries a direct relativistic signature on the breakdown velocity. Based on theoretical calculations of MLG- and BLG-ME spectra, we show that Doppler-induced instabilities manifest for a ME phase velocity determined by a universal value of the ME conductivity, set by the Hall conductance., Comment: 27 pages, 11 figures including supplementary information (14 pages and 3 figures for the main text alone)

Strong electric field annihilation by particle-antiparticle pair creation, also known as the Schwinger effect, is a non-perturbative prediction of quantum electrodynamics. Its experimental demonstration remains elusive, as threshold electric fields are extremely strong and beyond current reach. Here, we propose a mesoscopic variant of the Schwinger effect in graphene, which hosts Dirac fermions with an approximate electron-hole symmetry. Using transport measurements, we report on universal 1d-Schwinger conductance at the pinchoff of ballistic graphene transistors. Strong pinchoff electric fields are concentrated within approximately 1 $\mu$m of the transistor's drain, and induce Schwinger electron-hole pair creation at saturation. This effect precedes a collective instability toward an ohmic Zener regime, which is rejected at twice the pinchoff voltage in long devices. These observations advance our understanding of current saturation limits in ballistic graphene and provide a direction for further quantum electrodynamic experiments in the laboratory., Comment: 39 pages, 13 figures, final version with extended discussion of the pinchoff effect in supplementary informations, abstract updated

Optical response of crystalline solids is to a large extent driven by excitations that promote electrons among individual bands. This allows one to apply optical and magneto-optical methods to determine experimentally the energy band gap - a fundamental property crucial to our understanding of any solid - with a great precision. Here we show that such conventional methods, applied with great success to many materials in the past, do not work in topological Dirac semimetals with a dispersive nodal line. There, the optically deduced band gap depends on how the magnetic field is oriented with respect to the crystal axes. Such highly unusual behaviour is explained in terms of band-gap renormalization driven by Lorentz boosts., Comment: 9 pages, 3 figures + supplementary materials

Landau-level spectroscopy, the optical analysis of electrons in materials subject to a strong magnetic field, is a versatile probe of the electronic band structure and has been successfully used in the identification of novel states of matter such as Dirac electrons, topological materials or Weyl semimetals. The latter arise from a complex interplay between crystal symmetry, spin-orbit interaction and inverse ordering of electronic bands. Here, we report on unusual Landau-level transitions in the monopnictide TaP that decrease in energy with increasing magnetic field. We show that these transitions arise naturally at intermediate energies in time-reversal-invariant Weyl semimetals where the Weyl nodes are formed by a partially gapped nodal-loop in the band structure. We propose a simple theoretical model for electronic bands in these Weyl materials that captures the collected magneto-optical data to great extent., Comment: 6 pages, 3 figures + supplementary materials, accepted in Phys. Rev. Lett

We study the conductivity of two-dimensional graphene-type materials with tilted cones as well as their three-dimensional Weyl counterparts and show that a covariant quantum Boltzmann equation is capable of providing an accurate description of these materials' transport properties. The validity of the covariant Boltzmann approach is corroborated by calculations within the Kubo formula. We find a strong anisotropy in the conductivities parallel and perpendicular to the tilt direction upon increase of the tilt parameter $\eta$, which can be interpreted as the boost parameter of a Lorentz transformation. While the ratio between the two conductivities is $\sqrt{1-\eta^2}$ in the two-dimensional case where only the conductivity perpendicular to the tilt direction diverges for $\eta\rightarrow 1$, both conductivities diverge in three-dimensional Weyl semimetals, where $\eta=1$ separates a type-I (for $\eta<1$) from a type-II Weyl semimetal (for $\eta>1$)., Comment: 14 pages, 9 figures

In addition to the usual chiral surface states, massive surface states can arise at a smooth interface between a topological and a trivial bulk insulator. While not subject to topological protection as the chiral states, these massive states, theorized by Volkov and Pankratov in the 1980s, reflect nevertheless emergent Dirac physics at the interface. We study theoretically the magneto-optical response of these surface states, which is strikingly different from that of the bulk states. Most saliently, we show that these states can be identified clearly in the presence of a magnetic field and its orientation with respect to the interface., Comment: 5 pages with 2 figures, added references, fig.1a replaced

We theoretically study the role of the Berry curvature on neutral and charged excitons in two-dimensional transition-metal dichalcogenides. The Berry curvature arises due to a strong coupling between the conduction and valence bands in these materials that can to great extent be described within the model of massive Dirac fermions. The Berry curvature lifts the degeneracy of exciton states with opposite angular momentum. Using an electronic interaction that accounts for non-local screening effects, we find a Berry-curvature induced splitting of $\sim 17$ meV between the 2$p_{-}$ and 2$p_{+}$ exciton states in WS$_2$, consistent with experimental findings. Furthermore, we calculate the trion binding energies in WS$_2$ and WSe$_2$ for a large variety of screening lenghts and different dielectric constants for the environment. Our approach indicates the prominent role played by the Berry curvature along with non-local electronic interactions in the understanding of the energy spectra of neutral and charged excitons in transition-metal dichalcogenides and in the the interpretation of their optical properties., Comment: 11 pages, 3 figures

Breakdown of the quantum Hall effect (QHE) is commonly associated with an electric field approaching the inter Landau-level (LL) Zener field, ratio of the Landau gap and cyclotron radius. Eluded in semiconducting heterostructures, in spite of extensive investigation, the intrinsic Zener limit is reported here using high-mobility bilayer graphene and high-frequency current noise. We show that collective excitations arising from electron-electron interactions are essential. Beyond a noiseless ballistic QHE regime a large superpoissonian shot noise signals the breakdown via inter-LL scattering. The breakdown is ultimately limited by collective excitations in a regime where phonon and impurity scattering are quenched. The breakdown mechanism can be described by a Landau critical velocity as it bears strong similarities with the roton mechanism of superfluids., Comment: 13 pages, 3 figures

Cadmium arsenide (Cd3As2) has recently became conspicuous in solid-state physics due to several reports proposing that it hosts a pair of symmetry-protected 3D Dirac cones. Despite vast investigations, a solid experimental insight into the band structure of this material is still missing. Here we fill one of the existing gaps in our understanding of Cd3As2, and based on our Landau level spectroscopy study, we provide an estimate for the energy scale of 3D Dirac electrons in this system. We find that the appearance of such charge carriers is limited - contrary to a widespread belief in the solid-state community - to a relatively small energy scale (below 40 meV)., Comment: to be published in Phys. Rev. B

We report experimental measurements and theoretical analysis of Shubnikov-de Haas (SdH) oscillations in a Dirac cone system: the a-(BEDT-TTF)2I3 organic metal under hydrostatic pressure. The measured SdH oscillations reveal anomalies at high magnetic fields B where the 1/B oscillations periodicity is lost above 7 T. We interpret these unusual results within a theoretical model that takes into account intrinsic distortions of the a-(BEDT-TTF)2I3 Dirac cones such as a parabolic particle-hole asymmetric correction. Others possible causes, such as a cone tilting or a Zeeman effect, are carefully ruled out. The observations are consistent among a-(BEDT-TTF)2I3 samples with different Fermi levels.

We show that a smooth interface between two insulators of opposite topological Z2 indices possesses multiple surface states, both massless and massive. While the massless surface state is non-degenerate, chiral and insensitive to the interface potential, the massive surface states only appear for a sufficiently smooth heterojunction. The surface states are particle-hole symmetric and a voltage drop reveals their intrinsic relativistic nature, similarly to Landau bands of Dirac electrons in a magnetic field. We discuss the relevance of the massive Dirac surface states in recent ARPES and transport experiments., Comment: 5 pages, 3 figures; 6 pages of supplementary material incorporated; accepted for publication in Physical Review B as a Rapid Communication

We report on optical reflectivity experiments performed on Cd3As2 over a broad range of photon energies and magnetic fields. The observed response clearly indicates the presence of 3D massless charge carriers. The specific cyclotron resonance absorption in the quantum limit implies that we are probing massless Kane electrons rather than symmetry-protected 3D Dirac particles. The latter may appear at a smaller energy scale and are not directly observed in our infrared experiments., Comment: 5 pages, 4 figures + supplementary materials (17 pages), to be published in Phys. Rev. Lett

We have observed the quantum Hall effect (QHE) and Shubnikov-de Haas (SdH) oscillations in highly disordered graphene at magnetic fields up to 65 T. Disorder was introduced by hydrogenation of graphene up to a ratio H/C $\approx 0.1\%$. The analysis of SdH oscillations and QHE indicates that the topological part of the Berry phase, proportional to the pseudo-spin winding number, is robust against introduction of disorder by hydrogenation in large scale graphene.

We study the flat-band ferromagnetic phase of a topological Hubbard model within a bosonization formalism and, in particular, determine the spin-wave excitation spectrum. We consider a square lattice Hubbard model at 1/4-filling whose free-electron term is the \pi-flux model with topologically nontrivial and nearly flat energy bands. The electron spin is introduced such that the model either explicitly breaks time-reversal symmetry (correlated flat-band Chern insulator) or is invariant under time-reversal symmetry (correlated flat-band $Z_2$ topological insulator). We generalize for flat-band Chern and topological insulators the bosonization formalism [Phys. Rev. B 71, 045339 (2005)] previously developed for the two-dimensional electron gas in a uniform and perpendicular magnetic field at filling factor \nu=1. We show that, within the bosonization scheme, the topological Hubbard model is mapped into an effective interacting boson model. We consider the boson model at the harmonic approximation and show that, for the correlated Chern insulator, the spin-wave excitation spectrum is gapless while, for the correlated topological insulator, gapped. We briefly comment on the possible effects of the boson-boson (spin-wave--spin-wave) coupling., Comment: 16 pages, 5 figures

We investigate numerically different phases that can occur at half filling in the lowest and the first excited Landau levels in wide-well twodimensional electron systems exposed to a perpendicular magnetic field. Within a twocomponent model that takes into account only the two lowest electronic subbands of the quantum well, we derive a phase diagram that compares favorably with an experimental one by Shabani et al. [Phys. Rev. B 88, 245413 (2013)]. In addition to the compressible composite-fermion Fermi liquid in narrow wells with a substantial subband gap and the incompressible twocomponent (331) Halperin state, we identify in the lowest Landau level a rectangular Wigner crystal occupying the second subband. This crystal may be the origin of the experimentally observed insulating phase in the limit of wide wells and high electronic densities. In the second Landau level, the incompressible Pfaffian state, which occurs in narrow wells and large subband gaps, is also separated by an intermediate region from a large-well limit in which a similar rectangular Wigner crystal in the excited subband is the ground state, as for the lowest Landau level. However, the intermediate region is characterized by an incompressible state that consists of two four-flux Pfaffians in each of the components., Comment: 18 pages, 18 figures; published version with new appendix (discussion of the role of wave function overlap in the crystalline phases), updated references

We analyze the low-energy properties of two-dimensional direct-gap semiconductors, such as for example the transition-metal dichalcogenides MoS$_2$, WS$_2$, and their diselenide analogues MoSe$_2$, WSe$_2$, etc., which are currently intensively investigated. In general, their electrons have a mixed character -- they can be massive Dirac fermions as well as simple Schr\"odinger particles. We propose a measure (Diracness) for the degree of mixing between the two characters and discuss how this quantity can in principle be extracted experimentally, within magneto-transport measurements, and numerically via ab initio calculations., Comment: 6 pages, 2 figures ; new version (with minor modifications) accepted for publication in EPL

We study the fractional quantum Hall effect in a bilayer with charge-distribution imbalance induced, for instance, by a bias gate voltage. The bilayer can either be intrinsic or it can be formed spontaneously in wide quantum wells, due to the Coulomb repulsion between electrons. We focus on fractional quantum Hall effect in asymmetric bilayer systems at filling factor nu=4/11 and show that an asymmetric Halperin-like trial wavefunction gives a valid description of the ground state of the system., Comment: 6 pages, 5 figures, HMF-20 proceedings

We investigate theoretically the fractional quantum Hall effect at half-filling in the lowest Landau level observed in asymmetric wide quantum wells. The asymmetry can be achieved by a potential bias applied between the two sides of the well. Within exact-diagonalization calculations in the spherical geometry, we find that the incompressible state is described in terms of a two-component wave function. Its overlap with the ground state can be optimized with the help of a rotation in the space of the pseudospin, which mimics the lowest two electronic subbands., Comment: 6 pages, 6 figures; modified version contains discussion of energy gaps and results for another value of the magnetic field; version accepted for publication in Phys. Rev. B

We investigate the behavior of low-energy electrons in two-dimensional molybdenum disulfide when submitted to an external magnetic field. Highly degenerate Landau levels form in the material, between which light-induced excitations are possible. The dependence of excitations on light polarization and energy is explicitly determined, and it is shown that it is possible to induce valley and spin polarization, i.e. to excite electrons of selected valley and spin. Whereas the effective low-energy model in terms of massive Dirac fermions yields dipole-type selection rules, higher-order band corrections allow for the observation of additional transitions. Furthermore, inter-Landau-level transitions involving the n=0 levels provide a reliable method for an experimental measurement of the gap and the spin-orbit gap of molybdenum disulfide., Comment: 8 pages, 6 figures; published version with minor modifications

Transport measurements were performed on the organic layered compound \aI3 under hydrostatic pressure. The carrier types, densities and mobilities are determined from the magneto-conductance of \aI3 . While evidence of high-mobility massless Dirac carriers has already been given, we report here, their coexistence with low-mobility massive holes. This coexistence seems robust as it has been found up to the highest studied pressure. Our results are in agreement with recent DFT calculations of the band structure of this system under hydrostatic pressure. A comparison with graphene Dirac carriers has also been done., Comment: 5 pages 5 figures

We present infrared transmission spectroscopy study of the inter-Landau-level excitations in quasi-neutral epitaxial graphene nanoribbon arrays. We observed a substantial deviation in energy of the $L_{0(-1)}$$\to$$L_{1(0)}$ transition from the characteristic square root magnetic-field dependence of two-dimensional graphene. This deviation arises from the formation of upper-hybrid mode between the Landau level transition and the plasmon resonance. In the quantum regime the hybrid mode exhibits a distinct dispersion relation, markedly different from that expected for conventional two-dimensional systems and highly doped graphene.

The electrodynamics of a two-dimensional gas of massless fermions in graphene is studied by a collisionless hydrodynamic approach. A low-energy dispersion relation for the collective modes (plasmons) is derived both in the absence and in the presence of a perpendicular magnetic field. The results for graphene are compared to those for a standard two-dimensional gas of massive electrons. We further compare the results within the classical hydrodynamic approach to the full quantum mechanical calculation in the random phase approximation. The low-energy dispersion relation is shown to be a good approximation at small wave vectors. The limitations of this approach at higher order is also discussed., Comment: 7 pages, 1 figure

We investigate the fine structure in the energy spectrum of bilayer graphene in the presence of various stacking defaults, such as a translational or rotational mismatch. This fine structure consists of four Dirac points that move away from their original positions as a consequence of the mismatch and eventually merge in various manners. The different types of merging are described in terms of topological invariants (winding numbers) that determine the Landau-level spectrum in the presence of a magnetic field as well as the degeneracy of the levels. The Landau-level spectrum is, within a wide parameter range, well described by a semiclassical treatment that makes use of topological winding numbers. However, the latter need to be redefined at zero energy in the high-magnetic-field limit as well as in the vicinity of saddle points in the zero-field dispersion relation., Comment: 17 pages, 16 figures; published version with enhanced discussion of experimental findings

We review different scenarios for the motion and merging of Dirac points in two dimensional crystals. These different types of merging can be classified according to a winding number (a topological Berry phase) attached to each Dirac point. For each scenario, we calculate the Landau level spectrum and show that it can be quantitatively described by a semiclassical quantization rule for the constant energy areas. This quantization depends on how many Dirac points are enclosed by these areas. We also emphasize that different scenarios are characterized by different numbers of topologically protected zero energy Landau levels, Comment: 6 pages, 9 figures, Proceedings of ECRYS-2011, International School and Workshop on Electronic Crystals, Cargese (France)

We introduce a microscopic model on the honeycomb bilayer, which in the small-momentum limit captures the usual (quadratic dispersion in kinetic term) description of bilayer graphene. In the limit of strong interlayer hopping it reduces to an effective honeycomb monolayer model with also third neighbor hopping. We study interaction effects in this effective model focusing on possible superconducting instabilities. We find d_{x^2-y^2} superconductivity in the strong coupling limit of an effective tJ-model-like description that gradually transforms into d + id time-reversal symmetry breaking superconductivity at weak couplings. In this limit the small momentum order parameter expansion is (k_x + i k_y)^2 [or (k_x - ik_y)^2] in both valleys of the effective low-energy description. The relevance of our model and investigation for the physics of bilayer graphene is also discussed., Comment: 10 pages, published version

Inspired by the recent creation of the honeycomb optical lattice and the realization of the Mott insulating state in a square lattice by shaking, we study here the shaken honeycomb optical lattice. For a periodic shaking of the lattice, a Floquet theory may be applied to derive a time-independent Hamiltonian. In this effective description, the hopping parameters are renormalized by a Bessel function, which depends on the shaking direction, amplitude and frequency. Consequently, the hopping parameters can vanish and even change sign, in an anisotropic manner, thus yielding different band structures. Here, we study the merging and the alignment of Dirac points and dimensional crossovers from the two dimensional system to one dimensional chains and zero dimensional dimers. We also consider next-nearest-neighbor hopping, which breaks the particle-hole symmetry and leads to a metallic phase when it becomes dominant over the nearest-neighbor hopping. Furthermore, we include weak repulsive on-site interactions and find the density profiles for different values of the hopping parameters and interactions, both in a homogeneous system and in the presence of a trapping potential. Our results may be experimentally observed by using momentum-resolved Raman spectroscopy., Comment: 12 pages, 8 figures

We investigate the algebraic structure of flat energy bands a partial filling of which may give rise to a fractional quantum anomalous Hall effect (or a fractional Chern insulator) and a fractional quantum spin Hall effect. Both effects arise in the case of a sufficiently flat energy band as well as a roughly flat and homogeneous Berry curvature, such that the global Chern number, which is a topological invariant, may be associated with a local non-commutative geometry. This geometry is similar to the more familiar situation of the fractional quantum Hall effect in two-dimensional electron systems in a strong magnetic field., Comment: 8 pages, 3 figure; published version with labels in Figs. 2 and 3 corrected

We study the density of states in graphene at high magnetic field, when the physics is dominated by strong correlations between electrons. In particular we use the method of Haldane pseudopotentials to focus on almost empty or almost filled Landau levels. We find that, besides the usual Landau level peaks, additional peaks ("sashes") appear in the spectrum. The energies of these peaks are determined by the strength of Haldane's pseudopotentials, but as opposed to the usual two-dimensional gas, when there is a one-to-one correspondence between a Haldane pseudopotential and a peak in the spectrum, the energy of each peak is determined in general by a combination of more than one pseudopotential values. An eventual measure of these peak in the density of states spectrum of graphene would allow one to determine the value of the pseudopotentials in graphene, and thus test the strength of the interactions in this system., Comment: 9 pages, 1 table

We review the theoretical basis and understanding of electronic interactions in graphene Landau levels, in the limit of strong correlations. This limit occurs when inter-Landau-level excitations may be omitted because they belong to a high-energy sector, whereas the low-energy excitations only involve the same level, such that the kinetic energy (of the Landau level) is an unimportant constant. Two prominent effects emerge in this limit of strong electronic correlations: generalised quantum Hall ferromagnetic states that profit from the approximate four-fold spin-valley degeneracy of graphene's Landau levels and the fractional quantum Hall effect. Here, we discuss these effects in the framework of an SU(4)-symmetric theory, in comparison with available experimental observations., Comment: 12 pages, 3 figures; review for the proceedings of the Nobel Symposium on Graphene and Quantum Matter

This short theoretical review deals with some essential ingredients for the understanding of the quantum Hall effect in graphene in comparison with the effect in conventional two-dimensional electron systems with a parabolic band dispersion. The main difference between the two systems stems from the "ultra-relativistic" character of the low-energy carriers in graphene, which are described in terms of a Dirac equation, as compared to the non-relativistic Schr\"odinger equation used for electrons with a parabolic band dispersion. In spite of this fundamental difference, the Hall resistance quantisation is universal in the sense that it is given in terms of the universal constant h/e^2 and an integer number, regardless of whether the charge carriers are characterised by Galilean or Lorentz invariance, for non-relativistic or relativistic carriers, respectively., Comment: 9 pages, 4 figures; brief review article for Comptes Rendus de l'Academie des Sciences; references added with respect to previous version

We show that the twisted graphene bilayer can reveal unusual topological properties at low energies, as a consequence of a Dirac-point splitting. These features rely on a symmetry analysis of the electron hopping between the two layers of graphene and we derive a simplified effective low-energy Hamiltonian which captures the essential topological properties of twisted bilayer graphene. The corresponding Landau levels peculiarly reveal a degenerate zero-energy mode which cannot be lifted by strong magnetic fields., Comment: 5 pages, 3 figures; published version

We present a theoretical description of Bernstein modes that arise as a result of the coupling between plasmon-like collective excitations (upper-hybrid mode) and inter-Landau-level excitations, in graphene in a perpendicular magnetic field. These modes, which are apparent as avoided level crossings in the spectral function obtained in the random-phase approximation, are described to great accuracy in a phenomenological model. Bernstein modes, which may be measured in inelastic light-scattering experiments or in photo-conductivity spectroscopy, are a manifestation of the Coulomb interaction between the electrons and may be used for a high-precision measurement of the upper-hybrid mode at small non-zero wave vectors., Comment: 5 pages. 2 figures; slightly extended version, title changed; accepted for publication in Phys. Rev. B

We study collective electronic excitations in graphene in the integer quantum Hall regime, concentrating mainly on excitations with spin reversal such as spin-flip and spin-wave excitations. We show that these excitations are correctly accounted for in the time-dependent Hartree-Fock and strong magnetic field approximations, in contrast to spin-conserving (magneto-exciton) modes which involve a strong Landau-level mixing at non-zero wave vectors. The collective excitations are discussed in view of prominent theorems, such as Kohn's and Larmor's. Whereas the latter remains valid in graphene and yields insight into the understanding of spin-dependent modes, Kohn's theorem does not apply to relativistic electrons in graphene. We finally calculate the exchange correction to the chemical potential in the weak magnetic field limit., Comment: Replaced with published version. 13 pages, 4 figures

The semiclassical quantization of cyclotron orbits for two-dimensional Bloch electrons in a coupled two band model with a particle-hole symmetric spectrum is considered. As concrete examples, we study graphene (both mono and bilayer) and boron nitride. The main focus is on wave effects -- such as Berry phase and Maslov index -- occurring at order $\hbar$ in the semiclassical quantization and producing non-trivial shifts in the resulting Landau levels. Specifically, we show that the index shift appearing in the Landau levels is related to a topological part of the Berry phase -- which is basically a winding number of the direction of the pseudo-spin 1/2 associated to the coupled bands -- acquired by an electron during a cyclotron orbit and not to the complete Berry phase, as commonly stated. As a consequence, the Landau levels of a coupled band insulator are shifted as compared to a usual band insulator. We also study in detail the Berry curvature in the whole Brillouin zone on a specific example (boron nitride) and show that its computation requires care in defining the "k-dependent Hamiltonian" H(k), where k is the Bloch wavevector., Comment: 15 pages, 6 figures, submitted to EPJB

We study the recently observed graphene fractional quantum Hall state at a filling factor $\nu_G=1/3$ using a four-component trial wave function and exact diagonalization calculations. Although it is adiabatically connected to a 1/3 Laughlin state in the upper spin branch, with SU(2) valley-isospin ferromagnetic ordering and a completely filled lower spin branch, it reveals physical properties beyond such a state that is the natural ground state for a large Zeeman effect. Most saliently, it possesses at experimentally relevant values of the Zeeman gap low-energy spin-flip excitations that may be unveiled in inelastic light-scattering experiments., Comment: 4 pages, 3 figures; slightly modified published version

We review the basic aspects of electrons in graphene (two-dimensional graphite) exposed to a strong perpendicular magnetic field. One of its most salient features is the relativistic quantum Hall effect the observation of which has been the experimental breakthrough in identifying pseudo-relativistic massless charge carriers as the low-energy excitations in graphene. The effect may be understood in terms of Landau quantization for massless Dirac fermions, which is also the theoretical basis for the understanding of more involved phenomena due to electronic interactions. We present the role of electron-electron interactions both in the weak-coupling limit, where the electron-hole excitations are determined by collective modes, and in the strong-coupling regime of partially filled relativistic Landau levels. In the latter limit, exotic ferromagnetic phases and incompressible quantum liquids are expected to be at the origin of recently observed (fractional) quantum Hall states. Furthermore, we discuss briefly the electron-phonon coupling in a strong magnetic field. Although the present review has a dominating theoretical character, a close connection with available experimental observation is intended., Comment: 56 pages, 27 figures; published version with minor corrections and updated references

We examine the possibility of creating the Moore-Read Pfaffian in the lowest Landau level when the multicomponent Halperin 331 state (believed to describe quantum Hall bilayers and wide quantum wells at the filling factor $\nu=1/2$) is destroyed by the increase of tunneling. Using exact diagonalization of the bilayer Hamiltonian with short-range and long-range (Coulomb) interactions in spherical and periodic rectangular geometries, we establish that tunneling is a perturbation that drives the 331 state into a compressible composite Fermi liquid, with the possibility for an intermediate critical state that possesses some properties of the Moore-Read Pfaffian. These results are interpreted in the two-component BCS model for Cauchy pairing with a tunneling constraint. We comment on the conditions to be imposed on a system with fluctuating density in order to achieve the stable Pfaffian phase., Comment: 10 pages, 7 figures

The Chern-Simons approach has been widely used to explain fractional quantum Hall states in the framework of trial wave functions. In the present paper, we generalise the concept of Chern-Simons transformations to systems with any number of components (spin or pseudospin degrees of freedom), extending earlier results for systems with one or two components. We treat the density fluctuations by adding auxiliary gauge fields and appropriate constraints. The Hamiltonian is quadratic in these fields and hence can be treated as a harmonic oscillator Hamiltonian, with a ground state that is connected to the Halperin wave functions through the plasma analogy. We investigate several conditions on the coefficients of the Chern-Simons transformation and on the filling factors under which our model is valid. Furthermore, we discuss several singular cases, associated with symmetric states., Comment: 11 pages, shortened version, accepted for publication in Phys. Rev. B

The particle-hole excitation spectrum for doped graphene is calculated from the dynamical polarizability. We study the zero and finite magnetic field cases and compare them to the standard two-dimensional electron gas. The effects of electron-electron interaction are included within the random phase approximation. From the obtained polarizability, we study the screening effects and the collective excitations (plasmon, magneto-excitons, upper-hybrid mode and linear magneto-plasmons). We stress the differences with the usual 2DEG., Comment: Invited review article in Semicond. Sci. Technol.; accepted for publication

These lecture notes yield an introduction to quantum Hall effects both for non-relativistic electrons in conventional 2D electron gases (such as in semiconductor heterostructures) and relativistic electrons in graphene. After a brief historical overview in chapter 1, we discuss in detail the kinetic-energy quantisation of non-relativistic and the relativistic electrons in a strong magnetic field (chapter 2). Chapter 3 is devoted to the transport characteristics of the integer quantum Hall effect, and the basic aspects of the fractional quantum Hall effect are described in chapter 4. In chapter 5, we briefly discuss several multicomponent quantum Hall systems, namely the quantum Hall ferromagnetism, bilayer systems and graphene that may be viewed as a four-component system., Comment: 102 pages; lecture notes for the Singapore session ``Ultracold Gases and Quantum Information'' of Les Houches Summer School, 2009; v2 contains minor corrections and additional references

We propose a simple Hamiltonian to describe the motion and the merging of Dirac points in the electronic spectrum of two-dimensional electrons. This merging is a topological transition which separates a semi-metallic phase with two Dirac cones from an insulating phase with a gap. We calculate the density of states and the specific heat. The spectrum in a magnetic field B is related to the resolution of a Schrodinger equation in a double well potential. They obey the general scaling law e_n \propto B^{2/3} f_n(Delta /B^{2/3}. They evolve continuously from a sqrt{n B} to a linear (n+1/2)B dependence, with a [(n+1/2)B]^{2/3} dependence at the transition. The spectrum in the vicinity of the topological transition is very well described by a semiclassical quantization rule. This model describes continuously the coupling between valleys associated with the two Dirac points, when approaching the transition. It is applied to the tight-binding model of graphene and its generalization when one hopping parameter is varied. It remarkably reproduces the low field part of the Rammal-Hofstadter spectrum for the honeycomb lattice., Comment: 18 pages, 15 figures

We study under which general conditions a pair of Dirac points in the electronic spectrum of a two-dimensional crystal may merge into a single one. The merging signals a topological transition between a semi-metallic phase and a band insulator. We derive a universal Hamiltonian that describes the physical properties of the transition, which is controlled by a single parameter, and analyze the Landau-level spectrum in its vicinity. This merging may be observed in the organic salt alpha-(BEDT-TTF)_2 I_3 or in an optical lattice of cold atoms simulating deformed graphene., Comment: 4 pages, 5 figures

We investigate, with the help of Monte-Carlo and exact-diagonalization calculations in the spherical geometry, several compressible and incompressible candidate wave functions for the recently observed quantum Hall state at the filling factor $\nu=1/4$ in a wide quantum well. The quantum well is modeled as a two-component system by retaining its two lowest subbands. We make a direct connection with the phenomenological effective-bilayer model, which is commonly used in the description of a wide quantum well, and we compare our findings with the established results at $\nu=1/2$ in the lowest Landau level. At $\nu=1/4$, the overlap calculations for the Halperin (5,5,3) and (7,7,1) states, the generalized Haldane-Rezayi state and the Moore-Read Pfaffian, suggest that the incompressible state is likely to be realized in the interplay between the Halperin (5,5,3) state and the Moore-Read Pfaffian. Our numerics shows the latter to be very susceptible to changes in the interaction coefficients, thus indicating that the observed state is of multicomponent nature., Comment: 14 pages, 8 figures; minor changes, accepted for publication in Phys. Rev. B

Due to its fourfold spin-valley degeneracy, graphene in a strong magnetic field may be viewed as a four-component quantum Hall system. We investigate the consequences of this particular structure on a possible, yet unobserved, fractional quantum Hall effect in graphene within a trial-wavefunction approach and exact-diagonalisation calculations. This trial-wavefunction approach generalises an original idea by Halperin to account for the SU(2) spin in semiconductor heterostructures with a relatively weak Zeeman effect. Whereas the four-component structure at a filling factor nu=1/3 adds simply a SU(4)-ferromagnetic spinor ordering to the otherwise unaltered Laughlin state, the system favours a valley-unpolarised state at nu=2/5 and a completely unpolarised state at nu=4/9. Due to the similar behaviour of the interaction potential in the zero-energy graphene Landau level and the first excited one, we expect these states to be present in both levels., Comment: 6 pages, 1 figure; review article for Solid State Comm. (Graphene Week 2008)

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 find particular scaling properties of the density patterns if one fixes the ratio nu_N/M_e between the filling factor nu_N of the last partially filled Landau level and the number of electrons per bubble. Although the total density profile depends explicitly on M_e, the local density of states of the lowest peaks turns out to be identical regardless the number of electrons M_e. Whereas these electron-solid phases are reminiscent to those expected in the conventional two-dimensional electron gas in GaAs heterostructures in the quantum Hall regime, the local density of states and the scaling relations we highlight in this paper may be, in graphene, directly measured by spectroscopic means, such as e.g. scanning tunneling microscopy., Comment: 8 pages, 7 figures; minor corrections

The relativistic Landau levels in the layered organic material alpha-(BEDT-TTF)_2I_3 [BEDT-TTF=bis(ethylenedithio)tetrathiafulvalene] are sensitive to the tilt of the Dirac cones, which, as in the case of graphene, determine the low-energy electronic properties under appropriate pressure. We show that an applied inplane electric field, which happens to be in competition with the tilt of the cones, lifts the twofold valley degeneracy due to a different level spacing. The scenario may be tested in infrared transmission spectroscopy., Comment: 4 pages, 1 figure; version with minor corrections published in EPL

A doped graphene layer in the integer quantum Hall regime reveals a highly unusual particle-hole excitation spectrum, which is calculated from the dynamical polarizability in the random phase approximation. We find that the elementary neutral excitations in graphene in a magnetic field are unlike those of a standard two-dimensional electron gas (2DEG): in addition to the upper-hybrid mode, the particle-hole spectrum is reorganized in linear magneto-plasmons that disperse roughly parallel to $\omega=v_F q$, instead of the usual horizontal (almost dispersionless) magneto-excitons. These modes could be detected in an inelastic light scattering experiment., Comment: 8 pages, 3 figures. Version accepted for publication in Phys. Rev. B

We discuss charged topological spin textures in quantum Hall ferromagnets in which the electrons carry a pseudospin as well as the usual spin degree of freedom, as is the case in bilayer GaAs or monolayer graphene samples. We develop a theory which treats spin and pseudospin on a manifestly equal footing, which may also be of help in visualizing the relevant spin textures. We in particular consider the entanglement of spin and pseudospin in the presence of realistic anisotropies. An entanglement operator is introduced which generates families of degenerate Skyrmions with differing entanglement properties. We propose a local characterization of the latter, and touch on the role entangled Skyrmions play in the nuclear relaxation time of quantum Hall ferromagnets., Comment: 13 pages, 3 figures; discussion of collective skyrmion modes added; version accepted for publication in Phys. Rev. B

We propose a scheme to construct the most prominent Abelian and non-Abelian fractional quantum Hall states from K-component Halperin wave functions. In order to account for a one-component quantum Hall system, these SU(K) colors are distributed over all particles by an appropriate symmetrization. Numerical calculations corroborate the picture that the proposed scheme allows for a unification of both Abelian and non-Abelian trial wave functions in the study of one-component quantum Hall systems., Comment: 4 pages, 2 figures; revised version, published in Phys. Rev. Lett

We investigate a generalized two-dimensional Weyl Hamiltonian, which may describe the low-energy properties of mechanically deformed graphene and of the organic compound alpha-(BEDT-TTF)_2I_3 under pressure. The associated dispersion has generically the form of tilted anisotropic Dirac cones. The tilt arises due to next-nearest-neighbor hopping when the Dirac points, where the valence band touches the conduction band, do not coincide with crystallographic high-symmetry points within the first Brillouin zone. Within a semiclassical treatment, we describe the formation of Landau levels in a strong magnetic field, the relativistic form of which is reminiscent to that of graphene, with a renormalized Fermi velocity due to the tilt of the Dirac cones. These relativistic Landau levels, experimentally accessible via spectroscopy or even a quantum Hall effect measurement, may be used as a direct experimental verification of Dirac cones in alpha-(BEDT-TTF)_2I_3., Comment: 11 pages, 5 figures; version accepted for publication in PRB, contains a more detailed discussion of the zero-energy Landau level in the presence of the tilt, technical parts shifted to the appendix