Manifestations of topological effects in graphene

Graphene is a monoatomic layer of graphite with carbon atoms arranged in a two-dimensional honeycomb lattice configuration. It has been known for more than 60 years that the electronic structure of graphene can be modelled by two-dimensional massless relativistic fermions. This property gives rise to numerous applications, both in applied sciences and in theoretical physics. Electronic circuits made out of graphene could take advantage of its high electron mobility that is witnessed even at room temperature. In the theoretical domain the Dirac-like behaviour of graphene can simulate high energy effects, such as the relativistic Klein paradox. Even more surprisingly, topological effects can be encoded in graphene such as the generation of vortices, charge fractionalisation and the emergence of anyons. The impact of the topological effects on graphene's electronic properties can be elegantly described by the Atiyah–Singer index theorem. Here we present a pedagogical encounter of this theorem and review its various applications to graphene. A direct consequence of the index theorem is charge fractionalisation that is usually known from the fractional quantum Hall effect. The charge fractionalisation gives rise to the exciting possibility of realising graphene based anyons that unlike bosons or fermions exhibit fractional statistics. Besides being of theoretical interest, anyons are a strong candidate for performing error free quantum information processing.     

Fractional Quantum Conductance Plateaus in Mosaic‐Like Conductors and Their Similarities to the Fractional Quantum Hall Effect

A simple route to generate magnetotransport data is reported that results in fractional quantum Hall plateaus in the conductance without invoking strongly correlated physics. Ingredients to the generating model are conducting tiles with integer quantum Hall effect and metallic linkers, further Kirchhoff rules. When connecting few identical tiles in a mosaic, fractional steps occur in the conductance values. Richer spectra representing several fractions occur when the tiles are parametrically varied. Parts of the simulation data are supported with purposefully designed graphene mosaics in high magnetic fields. The findings emphasize that the occurrence of fractional conductance values, in particular in two‐terminal measurements, does not necessarily indicate interaction‐driven physics. The importance of an independent determination of charge densities is underscored and similarities with and differences to the fractional quantum Hall effect are critically discussed.     

Measurement of the ν=1/3 fractional quantum hall energy gap in suspended graphene

We report on magnetotransport measurements of multiterminal suspended graphene devices. Fully developed integer quantum Hall states appear in magnetic fields as low as 2 T. At higher fields the formation of longitudinal resistance minima and transverse resistance plateaus are seen corresponding to fractional quantum Hall states, most strongly for ν=1/3. By measuring the temperature dependence of these resistance minima, the energy gap for the 1/3 fractional state in graphene is determined to be at ～20 K at 14 T.     

石墨烯中新奇量子物态的研究

石墨烯特殊的晶格结构和能带结构赋予了它独特的电学性质. 近年来, 分数量子霍尔态、 魔角石墨烯中的 关联绝缘体态和超导态等现象的发现不断证明着石墨烯是一种理想的二维模型体系, 可用于实现一系列新奇的量子物态, 对石墨烯中新奇量子物态的探测和调控也一直是凝聚态物理领域的前沿研究热点之一. 本文将系统地介绍近年来石墨烯中对称性破缺量子物态的研究进展, 包括平带中强关联量子物态的研究以及谷赝自旋调控的研究, 并介绍一种在纳米尺度、 单电子精度上探测二维材料体系简并度及对称性破缺态的普适方法, 希望为相关领域的研究人员提供参考和借鉴.     

Topological approach to quantum Hall effects and its important applications: higher Landau levels,graphene and its bilayer

In this paper the topological approach to quantum Hall effects is carefully described. Commensurability conditions together with proposed generators of a system braid group are employed to establish the fractional quantum Hall effect hierarchies of conventional semiconductors, monolayer and bilayer graphene structures. Obtained filling factors are compared with experimental data and a very good agreement is achieved. Preliminary constructions of ground-state wave functions in the lowest Landau level are put forward. Furthermore, this work explains why pyramids of fillings from higher bands are not counterparts of the well-known composite-fermion hierarchy – it provides with the cause for an intriguing robustness of ν = 7/3 , 8/3 and 5/2 states (also in graphene). The argumentation why paired states can be developed in two-subband systems (wide quantum wells) only when the Fermi energy lies in the first Landau level is specified. Finally, the paper also clarifies how an additional surface in bilayer systems contributes to an observation of the fractional quantum Hall effect near half-filling, ν = 1/2 .     

Lattice-induced double-valley degeneracy lifting in graphene by a magnetic field

We show that the recently discovered double-valley splitting of the Landau levels in the quantum Hall effect in graphene can be explained as the perturbative orbital interaction of intravalley and intervalley microscopic orbital currents with a magnetic field. This effect is facilitated by the translationally noninvariant terms that correspond to graphene's crystallographic honeycomb symmetry but do not exist in the relativistic theory of massless Dirac fermions in quantum electrodynamics. We discuss recent data in view of these findings.     

Magneto-optics of strongly correlated two-dimensional electrons in single heterojunctions

Investigations of two-dimensional (2D) electron systems in semiconductors subjected to a strong perpendicular magnetic field with the use of photoluminescence are reviewed. The foundation of the optical spectroscopy method using the radiative recombination of 2D electrons with photoexcited holes bound to acceptors in a δ-doped monolayer in GaAs/Al _x Ga_1-x As single heterojunctions is presented. Optical spectroscopy studies of the energy spectra of 2D electrons imposed on transverse magnetic fields in the regimes of the integer and fractional quantum Hall effects are discussed. The relationship between the mean energy of the 2D electron gas and the first moment of their radiative recombination is analysed. It is shown that the magnetic field dependence of the first moment provides a method to measure the cyclotron, enhanced spin and quasiparticle energy gaps at the same time. Therefore it is shown how magneto-optics ‘see’ the ground state of interacting 2D electrons in the extreme quantum limit and how an optical ‘tool’ is efficient for the determination of Coulomb gaps of incompressible Fermi fluids in the fractional quantum Hall effect. Finally optical observations and studies of the Wigner crystallization of 2D electrons are presented. The corresponding liquid-solid phase diagram is discussed.     

Quantum Scars in Microwave Dielectric Photonic Graphene Billiards

In the band structure of graphene,the dispersion relation is linear around a Dirac point at the corners of the Brillouin zone.The closed graphene system has proven to be the ideal model to investigate relativistic quantum chaos phenomena.The electromagnetic material photonic graphene(PG) and electronic graphene not only have the same structural symmetry,but also have the similar band structure.Thus,we consider a stadium shaped resonant cavity filled with PG to demonstrate the relativistic quantum chaos phenomenon by numerical simulation.It is interesting that the relativistic quantum scars not only are identified in the PG cavities,but also appear and disappear repeatedly.The wave vector difference between repetitive scars on the same orbit is analyzed and confirmed to follow the quantization rule.The exploration will not only demonstrate a visual simulation of relativistic quantum scars but also propose a physical system for observing valley-dependent relativistic quantum scars,which is helpful for further understanding of quantum chaos.     

Two-dimensional topological insulator state and topological phase transition in bilayer graphene

We show that gated bilayer graphene hosts a strong topological insulator (TI) phase in the presence of Rashba spin-orbit (SO) coupling. We find that gated bilayer graphene under preserved time-reversal symmetry is a quantum valley Hall insulator for small Rashba SO coupling λ(R), and transitions to a strong TI when λ(R)>√[U(2)+t(⊥)(2)], where U and t(⊥) are, respectively, the interlayer potential and tunneling energy. Different from a conventional quantum spin Hall state, the edge modes of our strong TI phase exhibit both spin and valley filtering, and thus share the properties of both quantum spin Hall and quantum valley Hall insulators. The strong TI phase remains robust in the presence of weak graphene intrinsic SO coupling.     

The stability of the fractional quantum Hall effect in topological insulators

With the recent observation of graphene-like Landau levels at the surface of topological insulators, the possibility of fractional quantum Hall effect, which is a fundamental signature of strong correlations, has become of interest. Some experiments have reported intra-Landau level structure that is suggestive of fractional quantum Hall effect. This paper discusses the feasibility of fractional quantum Hall effect from a theoretical perspective, and argues that while this effect should occur, ideally, in the n=0 and |n|=1 Landau levels, it is ruled out in higher |n| Landau levels. Unlike graphene, the fractional quantum Hall effect in topological insulators is predicted to show an interesting asymmetry between n=1 and n=−1 Landau levels due to spin-orbit coupling.     

Fractional quantum Hall States of Dirac electrons in graphene

We have investigated the fractional quantum Hall states of Dirac electrons in a graphene layer in different Landau levels. The relativistic nature of the energy dispersion relation of electrons in graphene significantly modifies the interelectron interactions. This results in a specific dependence of the ground state energy and the energy gaps for electrons on the Landau-level index. For the valley-polarized states, i.e., at nu=1/m, m being an odd integer, the energy gaps have the largest values in the n=1 Landau level. For the valley-unpolarized states, e.g., for the 2/3 state, the energy gaps are suppressed for n=1 as compared to those at n=0. For both n=1 and n=0, the ground state of the 2/3 system is fully valley-unpolarized.     

Topological invariants for fractional quantum hall states

We calculate a topological invariant, whose value would coincide with the Chern number in the case of integer quantum Hall effect, for fractional quantum Hall states. In the case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In the case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot uniquely label distinct states.     

Anyon interpolation between exact integer and fractional quantum hall states in an electric field

Summary The exact perturbed Laughlin wave function in the presence of a quadratic central electric potential is deduced by the so-called adiabatic heuristic. The corresponding exact expression for the fractional quantum Hall current density, including electric-field inhomogeneity effects, is evaluated and shown to be related by a simple scaling transformation to that for the integer case. This leads to conjecture that the effect of disorder in the fractional case can be derived in general from that in the integer case. The author of this paper has agreed to not receive the proofs for correction.     

Topological spin Hall states, charged skyrmions, and superconductivity in two dimensions

We study the properties of two dimensional topological spin Hall insulators which arise through spontaneous breakdown of spin symmetry in systems that are spin rotation invariant. Such a phase breaks spin rotation but not time reversal symmetry and has a vector order parameter. Skyrmion configurations in this vector order parameter are shown to have an electric charge that is twice the electron charge. When the spin Hall order is destroyed by condensation of Skyrmions superconductivity results. This may happen either through doping or at fixed filling by tuning interactions to close the Skyrmion gap. In the latter case the superconductor-spin Hall insulator quantum phase transition can be second order even though the two phases break distinct symmetries.     

Multi-component fractional quantum Hall states in graphene: SU(4) versus SU(2)

Because of the spin and Dirac-valley degrees of freedom, graphene allows the observation of one-, two- or four-component fractional quantum Hall effects in different parameter regions. We address the stability of various states in the SU(2) and SU(4) limits. In the SU(4) limit, we predict that new low-energy Goldstone modes determine the stability of the fractional quantum Hall states at 2/5, 3/7, etc; SU(4) skyrmions are not found to be relevant for the low-energy physics. These results are discussed in light of experiments.