Evidence of Hofstadter's Fractal Energy Spectrum in the Quantized Hall Conductance

Spectroscopy of the fractal Hofstadter energy spectrum

Hofstadter's butterfly, the predicted energy spectrum for non-interacting electrons confined to a two-dimensional lattice in a magnetic field, is one of the most remarkable fractal structures in nature1. At rational ratios of magnetic flux quanta per lattice unit cell, this spectrum shows self-similar distributions of energy levels that reflect its recursive construction. For most materials, Hofstadter's butterfly is predicted under experimental conditions that are unachievable using laboratory-scale magnetic fields1-3. More recently, electrical transport studies have provided evidence for Hofstadter's butterfly in materials engineered to have artificially large lattice constants4-6, such as those with moiré superlattices7-10. Yet, so far, direct spectroscopy of the fractal energy spectrum predicted by Hofstadter nearly 50 years ago has remained out of reach. Here we use high-resolution scanning tunnelling microscopy/spectroscopy (STM/STS) to investigate the flat electronic bands in twisted bilayer graphene (TBG) near the predicted second magic angle11,12, an ideal setting for spectroscopic studies of Hofstadter's spectrum. Our study shows the fractionalization of flat moiré bands into discrete Hofstadter subbands and discerns experimental signatures of self-similarity of this spectrum. Moreover, our measurements uncover a spectrum that evolves dynamically with electron density, showing phenomena beyond that of Hofstadter's original model owing to the combined effects of strong correlations, Coulomb interactions and the quantum degeneracy of electrons in TBG.

Interplay of valley, layer and band topology towards interacting quantum phases in moiré bilayer graphene

In Bernal-stacked bilayer graphene (BBG), the Landau levels give rise to an intimate connection between valley and layer degrees of freedom. Adding a moiré superlattice potential enriches the BBG physics with the formation of topological minibands - potentially leading to tunable exotic quantum transport. Here, we present magnetotransport measurements of a high-quality bilayer graphene-hexagonal boron nitride (hBN) heterostructure. The zero-degree alignment generates a strong moiré superlattice potential for the electrons in BBG and the resulting Landau fan diagram of longitudinal and Hall resistance displays a Hofstadter butterfly pattern with a high level of detail. We demonstrate that the intricate relationship between valley and layer degrees of freedom controls the topology of moiré-induced bands, significantly influencing the energetics of interacting quantum phases in the BBG superlattice. We further observe signatures of field-induced correlated insulators, helical edge states and clear quantizations of interaction-driven topological quantum phases, such as symmetry broken Chern insulators.

Dimerized Hofstadter model in two-leg ladder quasi-crystals

We theoretically study topological features, band structure, and localization properties of a dimerized two-leg ladder with an oscillating on-site potential. The periodicity of the on-site potential can take either rational or irrational values. We consider two types of dimerized configurations; symmetric and asymmetric models. For rational values of the periodicity as long as inversion symmetry is preserved both symmetric and asymmetric ladders can host topological phases. Additionally, the energy spectrum of the models exhibits a fractal structure known as the Hofstadter butterfly spectrum, dependent on the dimerization of the hopping and the strength of the on-site potential. In the case of irrational values for the periodicity, a metal-insulator phase transition occurs with small values of the critical strength of the on-site potential in the dimerized cases. Our models incorporate the effects of lattice configuration and quasi-periodicity, paving the way for establishing platforms that host both topological and non-topological phase transitions.

Nonlinear Landau Fan Diagram for Graphene Electrons Exposed to a Moiré Potential

Due to Landau quantization, the conductance of two-dimensional electrons exposed to a perpendicular magnetic field exhibits oscillations that generate a fan of linear trajectories when plotted in the parameter space spanned by density and field. This fan looks identical, irrespective of the dispersion and field dependence of the Landau level energy. This is no surprise because the position of conductance minima depends solely on the level degeneracy that is linear in flux. The fractal energy spectrum that emerges within each Landau band when electrons are also exposed to a two-dimensional superlattice potential produces numerous additional oscillations, but they also create just linear fans for identical reasons. Here, we report conductance oscillations of graphene electrons exposed to a moiré potential that defy this general rule and form nonlinear trajectories in the density-field plane. We attribute this anomalous behavior to the simultaneous occupation of multiple minibands and magnetic breakdown-induced open orbits.

Reentrant Correlated Insulators in Twisted Bilayer Graphene at 25 T (2π Flux)

Twisted bilayer graphene (TBG) is remarkable for its topological flat bands, which drive strongly interacting physics at integer fillings, and its simple theoretical description facilitated by the Bistritzer-MacDonald Hamiltonian, a continuum model coupling two Dirac fermions. Because of the large moiré unit cell, TBG offers the unprecedented opportunity to observe reentrant Hofstadter phases in laboratory-strength magnetic fields near 25 T. This Letter is devoted to magic angle TBG at 2π flux where the magnetic translation group commutes. We use a newly developed gauge-invariant formalism to determine the exact single-particle band structure and topology. We find that the characteristic TBG flat bands reemerge at 2π flux, but, due to the magnetic field breaking C_{2z}T, they split and acquire Chern number ±1. We show that reentrant correlated insulating states appear at 2π flux driven by the Coulomb interaction at integer fillings, and we predict the characteristic Landau fans from their excitation spectrum.

Hofstadter butterfly and topological edge states in a quasiperiodic photonic crystal cavity array

Quasiperiodic structures with additional synthetic degrees of freedom have recently been recognized as a promising way for investigating high-dimensional topological phases with lower physical dimensions. Here, we investigated the well-known Harper-Aubry-André model on an integrated photonic platform by proposing a new design of a quasiperiodic photonic crystal (PhC) cavity array. This array is composed of closely coupled H1 PhC cavities with their cavity lengths being periodically modulated in the real space. The frequency spectrum of the structure shows the main features of the Hofstadter butterfly, which is one of the most important phenomena in the Harper-Aubry-André model. By varying the modulation phase, this structure exhibits nontrivial topology, which supports strongly localized topological edge states. These results have shown that quasiperiodic PhC cavity arrays can serve as the testbed for studying topological phases and new topological phenomena on an integrated platform.

Turbulence Hierarchy and Multifractality in the Integer Quantum Hall Transition

We offer a new perspective on the problem of characterizing mesoscopic fluctuations in the interplateau regions of the integer quantum Hall transition. We found that longitudinal and transverse conductance fluctuations, generated by varying the external magnetic field within a microscopic model, are multifractal and lead to distributions of conductance increments (magnetoconductance) with heavy tails (intermittency) and signatures of a hierarchical structure (cascade) in the corresponding stochastic process, akin to Kolmogorov's theory of fluid turbulence. We confirm this picture by interpreting the stochastic process of the conductance increments in the framework of H theory, which is a continuous-time stochastic approach that incorporates the basic features of Kolmogorov's theory. The multifractal analysis of the conductance "time series," combined with the H-theory formalism, provides strong support for the overall characterization of mesoscopic fluctuations in the quantum Hall transition as a multifractal stochastic phenomenon with multiscale hierarchy, intermittency, and cascade effects.

Band conductivity oscillations in a gate-tunable graphene superlattice

Electrons exposed to a two-dimensional (2D) periodic potential and a uniform, perpendicular magnetic field exhibit a fractal, self-similar energy spectrum known as the Hofstadter butterfly. Recently, related high-temperature quantum oscillations (Brown-Zak oscillations) were discovered in graphene moiré systems, whose origin lies in the repetitive occurrence of extended minibands/magnetic Bloch states at rational fractions of magnetic flux per unit cell giving rise to an increase in band conductivity. In this work, we report on the experimental observation of band conductivity oscillations in an electrostatically defined and gate-tunable graphene superlattice, which are governed both by the internal structure of the Hofstadter butterfly (Brown-Zak oscillations) and by a commensurability relation between the cyclotron radius of electrons and the superlattice period (Weiss oscillations). We obtain a complete, unified description of band conductivity oscillations in two-dimensional superlattices, yielding a detailed match between theory and experiment.

Suspended semiconductor nanostructures: physics and technology

The current state of research on quantum and ballistic electron transport in semiconductor nanostructures with a two-dimensional electron gas separated from the substrate and nanoelectromechanical systems is reviewed. These nanostructures fabricated using the surface nanomachining technique have certain unexpected features in comparison to their non-suspended counterparts, such as additional mechanical degrees of freedom, enhanced electron-electron interaction and weak heat sink. Moreover, their mechanical functionality can be used as an additional tool for studying the electron transport, complementary to the ordinary electrical measurements. The article includes a comprehensive review of spin-dependent electron transport and multichannel effects in suspended quantum point contacts, ballistic and adiabatic transport in suspended nanostructures, as well as investigations on nanoelectromechanical systems. We aim to provide an overview of the state-of-the-art in suspended semiconductor nanostructures and their applications in nanoelectronics, spintronics and emerging quantum technologies.

Universality of Hofstadter Butterflies on Hyperbolic Lattices

Motivated by recent realizations of hyperbolic lattices in superconducting waveguides and electric circuits, we compute the Hofstadter butterfly on regular hyperbolic tilings. Utilizing large hyperbolic lattices with periodic boundary conditions, we obtain the true bulk spectrum unaffected by boundary states. The butterfly spectrum with large extended gapped regions prevails, and its shape is universally determined by the fundamental tile, while the fractal structure is lost. We explain how these features originate from Landau levels in hyperbolic space and can be verified experimentally.

Unusual magnetotransport in twisted bilayer graphene

SignificanceWhen two sheets of graphene are twisted to the magic angle of 1.1∘, the resulting flat moiré bands can host exotic correlated electronic states such as superconductivity and ferromagnetism. Here, we show transport properties of a twisted bilayer graphene device at 1.38∘, far enough above the magic angle that we do not expect exotic correlated states. Instead, we see several unusual behaviors in the device's resistivity upon tuning both charge carrier density and perpendicular magnetic field. We can reproduce these behaviors with a surprisingly simple model based on Hofstadter's butterfly. These results shed light on the underlying properties of twisted bilayer graphene.

Topological Wannier cycles induced by sub-unit-cell artificial gauge flux in a sonic crystal

Gauge fields play a major role in understanding quantum effects. For example, gauge flux insertion into single unit cells is crucial towards detecting quantum phases and controlling quantum dynamics and classical waves. However, the potential of gauge fields in topological materials studies has not been fully exploited. Here, we experimentally demonstrate artificial gauge flux insertion into a single plaquette of a sonic crystal with a gauge phase ranging from 0 to 2π. We insert the gauge flux through a three-step process of dimensional extension, engineering a screw dislocation and dimensional reduction. Additionally, the single-plaquette gauge flux leads to cyclic spectral flows across multiple bandgaps that manifest as topological boundary states on the plaquette and emerge only when the flux-carrying plaquette encloses the Wannier centres. We termed this phenomenon as the topological Wannier cycle. This work paves the way towards sub-unit-cell gauge flux, enabling future studies on synthetic gauge fields and topological materials.

Hyperbolic band theory under magnetic field and Dirac cones on a higher genus surface

We explore the hyperbolic band theory under a magnetic field for the first time. Our theory is a general extension of the conventional band theory defined on a Euclidean lattice into the band theory on a general hyperbolic lattice/Riemann surface. Our methods and results can be confirmed experimentally by circuit quantum electrodynamics, which enables us to create novel materials in a hyperbolic space. To investigate the band structures, we construct directly the hyperbolic magnetic Bloch states and find that they form Dirac cones on a coordinate neighborhood. They can be regarded as a global quantum gravity solution detectable in a laboratory. Besides this is the first explicit example of a massless Dirac state on a higher genus surface. Moreover we show that the energy spectrum exhibits an unusual fractal structure refracting the negative curvature, when plotted as a function of a magnetic flux.

Hofstadter Topology: Noncrystalline Topological Materials at High Flux

The Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding ∼10^{4}  T magnetic fields to a trivial band structure. In this Letter, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. Remarkably, we find topological phases that cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with a nonzero Chern number or mirror Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with a nontrivial Kane-Mele invariant can be classified as a 3D topological insulator (TI) or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of twofold rotation and time reversal and show that there exists a higher order TI phase where corner modes are pumped by flux. We show that a model of twisted bilayer graphene realizes this phase. Our results rely primarily on the magnetic translation group that exists at rational values of the flux. The advent of Moiré lattices renders our work relevant experimentally. Due to the enlarged Moiré unit cell, it is possible for laboratory-strength fields to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.

Gate-Tunable Two-Dimensional Superlattices in Graphene

We report an efficient technique to induce gate-tunable two-dimensional superlattices in graphene by the combined action of a back gate and a few-layer graphene patterned bottom gate complementary to existing methods. The patterned gates in our approach can be easily fabricated and implemented in van der Waals stacking procedures, allowing flexible use of superlattices with arbitrary geometry. In transport measurements on a superlattice with a lattice constant a = 40 nm, well-pronounced satellite Dirac points and signatures of the Hofstadter butterfly including a nonmonotonic quantum Hall response are observed. Furthermore, the experimental results are accurately reproduced in transport simulations and show good agreement with features in the calculated band structure. Overall, we present a comprehensive picture of graphene-based superlattices, featuring a broad range of miniband effects, both in experiment and in theoretical modeling. The presented technique is suitable for studying more advanced geometries which are not accessible by other methods.

Generation of Hofstadter's butterfly spectrum using circular arrays of microring resonators

Hofstadter's butterfly spectrum, which characterizes the energy bands of electrons in a 2D lattice under a perpendicular magnetic field, has been emulated and experimentally characterized in periodic bandgap structures at microwave and acoustic frequencies. However, measurement of the complete spectrum at optical frequencies has yet to be demonstrated. Here, we propose a simple topological photonic structure based on a circular array of microrings with periodic resonant frequency detunings that can be implemented on an integrated optics platform. We show that this ring-of-rings structure exactly emulates the Harper equation and propose an experimental approach for measuring Hofstadter's butterfly spectrum at optical frequencies.

Staggered potential and magnetic field tunable electronic switch in a kagome nanoribbon junction

We propose a possible electronic switch on a two dimensional (2D) kagome lattice by applying a perpendicular inhomogeneous magnetic field and a staggered sublattice potential. By means of the tight-binding lattice model and the non-equilibrium Green's function method, we calculate the quantum Hall conductance of the device at zero temperature limit. The numerical results demonstrate that when a staggered lattice potential is considered, the conventional integer Hall effect is changed into discrete fractional conductance peaks, and a finite energy gap can be opened in the system, which may induce a metal-insulator transition and can be designed as a 2D electronic valve. The conductance valve phenomena mainly come from the interplay between the asymmetry energy band induced by the magnetic field and a band gap opened by the staggered potential. The ON(OFF) state of the electron transport is efficiently controlled by the device parameters such as the magnetic field, the staggered lattice potential and the Fermi level. Our findings might be useful for designing efficient current valves in 2D nano-electronic devices.

New Generation of Moiré Superlattices in Doubly Aligned hBN/Graphene/hBN Heterostructures

The specific rotational alignment of two-dimensional lattices results in a moiré superlattice with a larger period than the original lattices and allows one to engineer the electronic band structure of such materials. So far, transport signatures of such superlattices have been reported for graphene/hBN and graphene/graphene systems. Here we report moiré superlattices in fully hBN encapsulated graphene with both the top and the bottom hBN aligned to the graphene. In the graphene, two different moiré superlattices form with the top and the bottom hBN, respectively. The overlay of the two superlattices can result in a third superlattice with a period larger than the maximum period (14 nm) in the graphene/hBN system, which we explain in a simple model. This new type of band structure engineering allows one to artificially create an even wider spectrum of electronic properties in two-dimensional materials.

Band structure engineering of 2D materials using patterned dielectric superlattices

The ability to manipulate electrons in two-dimensional materials with external electric fields provides a route to synthetic band engineering. By imposing artificially designed and spatially periodic superlattice potentials, electronic properties can be further altered beyond the constraints of naturally occurring atomic crystals^1-5. Here, we report a new approach to fabricate high-mobility superlattice devices by integrating surface dielectric patterning with atomically thin van der Waals materials. By separating the device assembly and superlattice fabrication processes, we address the intractable trade-off between device processing and mobility degradation that constrains superlattice engineering in conventional systems. The improved electrostatics of atomically thin materials allows smaller wavelength superlattice patterns relative to previous demonstrations. Moreover, we observe the formation of replica Dirac cones in ballistic graphene devices with sub-40 nm wavelength superlattices and report fractal Hofstadter spectra^6-8 under large magnetic fields from superlattices with designed lattice symmetries that differ from that of the host crystal. Our results establish a robust and versatile technique for band structure engineering of graphene and related van der Waals materials with dynamic tunability.

High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices

Cyclotron motion of charge carriers in metals and semiconductors leads to Landau quantization and magneto-oscillatory behavior in their properties. Cryogenic temperatures are usually required to observe these oscillations. We show that graphene superlattices support a different type of quantum oscillation that does not rely on Landau quantization. The oscillations are extremely robust and persist well above room temperature in magnetic fields of only a few tesla. We attribute this phenomenon to repetitive changes in the electronic structure of superlattices such that charge carriers experience effectively no magnetic field at simple fractions of the flux quantum per superlattice unit cell. Our work hints at unexplored physics in Hofstadter butterfly systems at high temperatures.

Observation of Dirac bands in artificial graphene in small-period nanopatterned GaAs quantum wells

Charge carriers in graphene behave like massless Dirac fermions (MDFs) with linear energy-momentum dispersion ^{1, 2} , providing a condensed-matter platform for studying quasiparticles with relativistic-like features. Artificial graphene (AG)-a structure with an artificial honeycomb lattice-exhibits novel phenomena due to the tunable interplay between topology and quasiparticle interactions ^3-6 . So far, the emergence of a Dirac band structure supporting MDFs has been observed in AG using molecular ⁵ , atomic ^{6, 7} and photonic systems ^8-10 , including those with semiconductor microcavities ¹¹ . Here, we report the realization of an AG that has a band structure with vanishing density of states consistent with the presence of MDFs. This observation is enabled by a very small lattice constant (a = 50 nm) of the nanofabricated AG patterns superimposed on a two-dimensional electron gas hosted by a high-quality GaAs quantum well. Resonant inelastic light-scattering spectra reveal low-lying transitions that are not present in the unpatterned GaAs quantum well. These excitations reveal the energy dependence of the joint density of states for AG band transitions. Fermi level tuning through the Dirac point results in a collapse of the density of states at low transition energy, suggesting the emergence of the MDF linear dispersion in the AG.

Spectral Flow and Global Topology of the Hofstadter Butterfly

We study the relation between the global topology of the Hofstadter butterfly of a multiband insulator and the topological invariants of the underlying Hamiltonian. The global topology of the butterfly, i.e., the displacement of the energy gaps as the magnetic field is varied by one flux quantum, is determined by the spectral flow of energy eigenstates crossing gaps as the field is tuned. We find that for each gap this spectral flow is equal to the topological invariant of the gap, i.e., the net number of edge modes traversing the gap. For periodically driven systems, our results apply to the spectrum of quasienergies. In this case, the spectral flow of the sum of all the quasienergies gives directly the Rudner-Lindner-Berg-Levin invariant that characterizes the topological phases of a periodically driven system.

Recent progress in van der Waals heterojunctions

Following the development of many novel two-dimensional (2D) materials, investigations of van der Waals heterojunctions (vdWHs) have attracted significant attention due to their excellent properties such as smooth heterointerface, highly gate-tunable bandgap, and ultrafast carrier transport. Benefits from the atom-scale thickness, physical and chemical properties and ease of manipulation of the heterojunctions formulated by weak vdW forces were demonstrated to indicate their outstanding potential in electronic and optoelectronic applications, including photodetection and energy harvesting, and the possibility of integrating them with the existing semiconductor technology for the next-generation electronic and sensing devices. In this review, we summarized the recent developments of vdWHs and emphasized their applications. Basically, we introduced the physical properties and some newly discovered phenomena in vdWHs. Then, we emphatically presented four classical vdWHs and some novel heterostructures formed by vdW forces. Based on their unique physical properties and structures, we highlighted the applications of vdWHs including in photodiodes, phototransistors, tunneling devices, and memory devices. Finally, we provided a conclusion on the recent advances in vdWHs and outlined our perspectives. We aim for this review to serve as a solid foundation in this field and to pave the way for future research on vdW-based materials and their heterostructures.

Hofstadter butterfly in the Falicov-Kimball model on some finite 2D lattices

Spinless, interacting electrons on a finite size triangular lattice moving in an extremely strong perpendicular magnetic field are studied in comparison to a square lattice. Using a Falicov-Kimball model, the effects of Coulomb correlation, magnetic field and finite system size on their energy spectrum are observed. Exact diagonalization and Monte Carlo simulation methods (based on a modified Metropolis algorithm) have been employed to examine the recursive structure of the Hofstadter spectrum in the presence of several electronic correlation strengths for different system sizes. It is possible to introduce a gap in the density of states even in the absence of electron correlation, which is anticipated as a metal to insulator transition driven by an orbital magnetic field. With further inclusion of the interaction, the gap in the spectrum is modified and in some cases the correlation is found to suppress extra states manifested by the finite size effects. At a certain flux, the opened gap due to magnetic field is reduced by the Coulomb interaction. An orbital current is calculated for both the square and the triangular lattice with and without electron correlation. In the non-interacting limit, the bulk current shows several patterns, while the edge current shows oscillations with magnetic flux. The oscillations persist in the interacting limit for the square lattice, but not for the triangular lattice.

Quantum Hall effect in epitaxial graphene with permanent magnets

We have observed the well-kown quantum Hall effect (QHE) in epitaxial graphene grown on silicon carbide (SiC) by using, for the first time, only commercial NdFeB permanent magnets at low temperature. The relatively large and homogeneous magnetic field generated by the magnets, together with the high quality of the epitaxial graphene films, enables the formation of well-developed quantum Hall states at Landau level filling factors v = ±2, commonly observed with superconducting electro-magnets. Furthermore, the chirality of the QHE edge channels can be changed by a top gate. These results demonstrate that basic QHE physics are experimentally accessible in graphene for a fraction of the price of conventional setups using superconducting magnets, which greatly increases the potential of the QHE in graphene for research and applications.

Commensurability Oscillations of Composite Fermions Induced by the Periodic Potential of a Wigner Crystal

When the kinetic energy of a collection of interacting two-dimensional (2D) electrons is quenched at very high magnetic fields so that the Coulomb repulsion dominates, the electrons are expected to condense into an ordered array, forming a quantum Wigner crystal (WC). Although this exotic state has long been suspected in high-mobility 2D electron systems at very low Landau level fillings (ν≪1), its direct observation has been elusive. Here we present a new technique and experimental results directly probing the magnetic-field-induced WC. We measure the magnetoresistance of a bilayer electron system where one layer has a very low density and is in the WC regime (ν≪1), while the other ("probe") layer is near ν=1/2 and hosts a sea of composite fermions (CFs). The data exhibit commensurability oscillations in the magnetoresistance of the CF layer, induced by the periodic potential of WC electrons in the other layer, and provide a unique, direct glimpse at the symmetry of the WC, its lattice constant, and melting. They also demonstrate a striking example of how one can probe an exotic many-body state of 2D electrons using equally exotic quasiparticles of another many-body state.

Artificial Gauge Field and Topological Phase in a Conventional Two-dimensional Electron Gas with Antidot Lattices

Based on the Born-Oppemheimer approximation, we divide the total electron Hamiltonian in a spin-orbit coupled system into the slow orbital motion and the fast interband transition processes. We find that the fast motion induces a gauge field on the slow orbital motion, perpendicular to the electron momentum, inducing a topological phase. From this general designing principle, we present a theory for generating artificial gauge field and topological phase in a conventional two-dimensional electron gas embedded in parabolically graded GaAs/InxGa1-xAs/GaAs quantum wells with antidot lattices. By tuning the etching depth and period of the antidot lattices, the band folding caused by the antidot potential leads to the formation of minibands and band inversions between neighboring subbands. The intersubband spin-orbit interaction opens considerably large nontrivial minigaps and leads to many pairs of helical edge states in these gaps.

Fractional quantum Hall effect in Hofstadter butterflies of Dirac fermions

We report on the influence of a periodic potential on the fractional quantum Hall effect (FQHE) states in monolayer graphene. We have shown that for two values of the magnetic flux per unit cell (one-half and one-third flux quantum) an increase of the periodic potential strength results in a closure of the FQHE gap and appearance of gaps due to the periodic potential. In the case of one-half flux quantum this causes a change of the ground state and consequently the change of the momentum of the system in the ground state. While there is also crossing between low-lying energy levels for one-third flux quantum, the ground state does not change with the increase of the periodic potential strength and is always characterized by the same momentum. Finally, it is shown that for one-half flux quantum the emergent gaps are due entirely to the electron-electron interaction, whereas for the one-third flux quantum per unit cell these are due to both non-interacting electrons (Hofstadter butterfly pattern) and the electron-electron interaction.

Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems

We present the science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems, targeting an evolution in technology, that might lead to impacts and benefits reaching into most areas of society. This roadmap was developed within the framework of the European Graphene Flagship and outlines the main targets and research areas as best understood at the start of this ambitious project. We provide an overview of the key aspects of graphene and related materials (GRMs), ranging from fundamental research challenges to a variety of applications in a large number of sectors, highlighting the steps necessary to take GRMs from a state of raw potential to a point where they might revolutionize multiple industries. We also define an extensive list of acronyms in an effort to standardize the nomenclature in this emerging field.

Gap structure of the Hofstadter system of interacting Dirac fermions in graphene

The effects of mutual Coulomb interactions between Dirac fermions in monolayer graphene on the Hofstadter energy spectrum are investigated. For two flux quanta per unit cell of the periodic potential, interactions open a gap in each Landau level with the smallest gap in the n=1 Landau level. For more flux quanta through the unit cell, where the noninteracting energy spectra have many gaps in each Landau level, interactions enhance the low-energy gaps and strongly suppress the high-energy gaps and almost close a high-energy gap for n=1. The signature of the interaction effects in the Hofstadter system can be probed through magnetization, which is governed by the mixing of the Landau levels and is enhanced by the Coulomb interaction.

Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices

We experimentally implement the Harper Hamiltonian for neutral particles in optical lattices using laser-assisted tunneling and a potential energy gradient provided by gravity or magnetic field gradients. This Hamiltonian describes the motion of charged particles in strong magnetic fields. Laser-assisted tunneling processes are characterized by studying the expansion of the atoms in the lattice. The band structure of this Hamiltonian should display Hofstadter's butterfly. For fermions, this scheme should realize the quantum Hall effect and chiral edge states.

Artificial honeycomb lattices for electrons, atoms and photons

Artificial honeycomb lattices offer a tunable platform for studying massless Dirac quasiparticles and their topological and correlated phases. Here we review recent progress in the design and fabrication of such synthetic structures focusing on nanopatterning of two-dimensional electron gases in semiconductors, molecule-by-molecule assembly by scanning probe methods and optical trapping of ultracold atoms in crystals of light. We also discuss photonic crystals with Dirac cone dispersion and topologically protected edge states. We emphasize how the interplay between single-particle band-structure engineering and cooperative effects leads to spectacular manifestations in tunnelling and optical spectroscopies.

Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure

van der Waals heterostructures constitute a new class of artificial materials formed by stacking atomically thin planar crystals. We demonstrated band structure engineering in a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally aligned hexagonal boron nitride substrate. The spatially varying interlayer atomic registry results in both a local breaking of the carbon sublattice symmetry and a long-range moiré superlattice potential in the graphene. In our samples, this interplay between short- and long-wavelength effects resulted in a band structure described by isolated superlattice minibands and an unexpectedly large band gap at charge neutrality. This picture is confirmed by our observation of fractional quantum Hall states at ± 5/3 filling and features associated with the Hofstadter butterfly at ultrahigh magnetic fields.

Hofstadter's butterfly and the fractal quantum Hall effect in moiré superlattices

Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum. Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely. Here we demonstrate that moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.

Cloning of Dirac fermions in graphene superlattices

Superlattices have attracted great interest because their use may make it possible to modify the spectra of two-dimensional electron systems and, ultimately, create materials with tailored electronic properties. In previous studies (see, for example, refs 1-8), it proved difficult to realize superlattices with short periodicities and weak disorder, and most of their observed features could be explained in terms of cyclotron orbits commensurate with the superlattice. Evidence for the formation of superlattice minibands (forming a fractal spectrum known as Hofstadter's butterfly) has been limited to the observation of new low-field oscillations and an internal structure within Landau levels. Here we report transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions. The substrate's moiré potential acts as a superlattice and leads to profound changes in the graphene's electronic spectrum. Second-generation Dirac points appear as pronounced peaks in resistivity, accompanied by reversal of the Hall effect. The latter indicates that the effective sign of the charge carriers changes within graphene's conduction and valence bands. Strong magnetic fields lead to Zak-type cloning of the third generation of Dirac points, which are observed as numerous neutrality points in fields where a unit fraction of the flux quantum pierces the superlattice unit cell. Graphene superlattices such as this one provide a way of studying the rich physics expected in incommensurable quantum systems and illustrate the possibility of controllably modifying the electronic spectra of two-dimensional atomic crystals by varying their crystallographic alignment within van der Waals heterostuctures.

Hall conductance in graphene with point defects

We investigate the Hall conductance of graphene with point defects within the Kubo formalism, which allows us to calculate the Hall conductance without constraining the Fermi energy to lie in a gap. For pure graphene, which we model using a tight-binding Hamiltonian, we recover both the usual and the anomalous integer quantum Hall effects depending on the proximity to the Dirac points. We investigate the effect of point defects on Hall conduction by considering a dilute but regular array of point defects incorporated into the graphene lattice. We extend our calculations to include next nearest neighbor hopping, which breaks the bipartite symmetry of the lattice. We find that impurity atoms which are weakly coupled to the rest of the lattice result in gradual disappearance of the high conductance value plateaus. For such impurities, especially for vacancies which are decoupled from the lattice, strong modification of the Hall conductance occurs near the E = 0 eV line, as impurity states are highly localized. In contrast, if the impurities are strongly coupled, they create additional Hall conductance plateaus at the extremum values of the spectrum, signifying separate impurity bands. Hall conductance values within the original spectrum are not strongly modified.

The integer quantum Hall effect of a square lattice with an array of point defects

The electronic properties of a square lattice under an applied perpendicular magnetic field in the presence of impurities or vacancies are investigated by the tight-binding method including up to second nearest neighbor interactions. These imperfections result in new gaps and bands in the Hofstadter butterfly even when the second order interactions break the bipartite symmetry. In addition, the whole spectrum of the Hall conduction is obtained by the Kubo formula for the corresponding cases. The results are in accordance with the Thouless-Kohmoto-Nightingale-den Nijs integers when the Fermi energy lies in an energy gap. We find that the states due to the vacancies or impurities with small hopping constants are highly localized and do not contribute to the Hall conduction. However, the impurities with high hopping constants result in new Hall plateaus with constant conduction of σ(xy) =± e(2)/h, since high hopping constants increase the probability of an electron contributing to the conduction.

Fractal Landau-level spectra in twisted bilayer graphene

The Hofstadter butterfly spectrum for Landau levels in a two-dimensional periodic lattice is a rare example exhibiting fractal properties in a truly quantum system. However, the observation of this physical phenomenon in a conventional material will require a magnetic field strength several orders of magnitude larger than what can be produced in a modern laboratory. It turns out that for a specific range of rotational angles twisted bilayer graphene serves as a special system with a fractal energy spectrum under laboratory accessible magnetic field strengths. This unique feature arises from an intriguing electronic structure induced by the interlayer coupling. Using a recursive tight-binding method, we systematically map out the spectra of these Landau levels as a function of the rotational angle. Our results give a complete description of LLs in twisted bilayer graphene for both commensurate and incommensurate rotational angles and provide quantitative predictions of magnetic field strengths for observing the fractal spectra in these graphene systems.

Magnetic modulation of the tunnelling between defect states in antidot superlattices

We show theoretically that the tunnelling between properly designed defects in periodic antidot lattices can be strongly modulated by applied magnetic fields. Further, transport channels made up of linear arrangements of tunnel-coupled defects can accommodate Aharonov-Bohm cages, suggesting a magnetic control of the transport through the system. Evidence supporting an unusual robustness of the caging effect against electron-electron interactions is also provided.

Effects of the Rashba spin-orbit coupling on Hofstadter's butterfly

We study the effect of Rashba spin-orbit coupling on the Hofstadter spectrum of a two-dimensional tight-binding electron system in a perpendicular magnetic field. We obtain the generalized coupled Harper spin-dependent equations which include the Rashba spin-orbit interaction and solve for the energy spectrum and spin polarization. We investigate the effect of spin-orbit coupling on the fractal energy spectrum and the spin polarization for some characteristic states as a function of the magnetic flux α and the spin-orbit coupling parameter. We characterize the complexity of the fractal geometry of the spin-dependent Hofstadter butterfly with the correlation dimension and show that it grows quadratically with the amplitude of the spin-orbit coupling. We study some ground state properties and the spin polarization shows a fractal-like behavior as a function of α, which is demonstrated with the exponent close to unity of the decaying power spectrum of the spin polarization. Some degree of spin localization or distribution around +1 or -1, for small spin-orbit coupling, is found with the determination of the entropy function as a function of the spin-orbit coupling. The excited states show a more extended (uniform) distribution of spin states.

Trapped Fermi gases in rotating optical lattices: realization and detection of the topological hofstadter insulator

We consider a gas of noninteracting spinless fermions in a rotating optical lattice and calculate the density profile of the gas in an external confinement potential. The density profile exhibits distinct plateaus, which correspond to gaps in the single particle spectrum known as the Hofstadter butterfly. The plateaus result from insulating behavior whenever the Fermi energy lies within a gap. We discuss the necessary conditions to realize the Hofstadter insulator in a cold atom setup and show how the quantized Hall conductance can be measured from density profiles using the Streda formula.

High-frequency magneto-oscillations in GaAs/AlGaAs quantum wells

We have observed very high-frequency, highly reproducible magneto-oscillations in modulation doped GaAs/AlGaAs quantum well structures. The oscillations are periodic in an inverse magnetic field (1/B) and their amplitude increases with temperature up to T approximately 700 mK. Being initially most pronounced around the filling factor nu=1/2, they move towards lower nu with increasing T. Front and back-gating data imply that these oscillations require a coupling to a parallel conducting layer. A comparison with existing oscillation models renders no explanation.

Photoconductivity in AC-driven modulated two-dimensional electron gas in a perpendicular magnetic field

In this work we study the microwave photoconductivity of a two-dimensional electron system (2DES) in the presence of a magnetic field and a two-dimensional modulation (2D). The model includes the microwave and Landau contributions in a non-perturbative exact way; the periodic potential is treated perturbatively. The Landau-Floquet states provide a convenient base with respect to which the lattice potential becomes time dependent, inducing transitions between the Landau-Floquet levels. Based on this formalism, we provide a Kubo-like formula that takes into account the oscillatory Floquet structure of the problem. The total longitudinal conductivity and resistivity exhibit strong oscillations, determined by ϵ = ω/ω(c), with ω the radiation frequency and ω(c) the cyclotron frequency. The oscillations follow a pattern with minima centred at [Formula: see text], and maxima centred at [Formula: see text], where j = 1,2,3..., δ∼1/5 is a constant shift and l is the dominant multipole contribution. Negative resistance states (NRSs) develop as the electron mobility and the intensity of the microwave power are increased. These NRSs appear in a narrow window region of values of the lattice parameter (a), around a∼l(B), where l(B) is the magnetic length. It is proposed that these phenomena may be observed in artificially fabricated arrays of periodic scatterers at the interface of ultraclean GaAs /Al(x)Ga(1-x)As heterostructures.

Manipulating spin and charge in magnetic semiconductors using superconducting vortices

The continuous need for miniaturization and increase in device speed drives the electronics industry to explore new avenues of information processing. One possibility is to use electron spin to store, manipulate and carry information. All such 'spintronics' applications are faced with formidable challenges in finding fast and efficient ways to create, transport, detect, control and manipulate spin textures and currents. Here we show how most of these operations can be performed in a relatively simple manner in a hybrid system consisting of a superconducting film and a paramagnetic diluted magnetic semiconductor (DMS) quantum well. Our proposal is based on the observation that the inhomogeneous magnetic fields of the superconducting film create local spin and charge textures in the DMS quantum well, leading to a variety of effects such as Bloch oscillations and an unusual quantum Hall effect. We exploit recent progress in manipulating magnetic flux bundles (vortices) in superconductors and show how these can create, manipulate and control the spin textures in DMSs.

Statistics of spectra for critical quantum chaos in one-dimensional quasiperiodic systems

We study spectral statistics of one-dimensional quasiperiodic systems at the metal-insulator transition. Several types of spectral statistics are observed at the critical points, lines, and region. On the critical lines, we find the bandwidth distribution P(B) (w) around the origin (in the tail) to have the form of P(B) (w) approximately w(alpha) [P(B) (w) approximately e(-beta w(gamma) ) ] (alpha,beta,gamma >0) , while in the critical region P(B) (w) approximately w(- alpha') (alpha' >0) . We also find the level spacing distribution to follow an inverse power law P(G) (s) approximately s(-delta) (delta>0) .

Detection of a Landau band-coupling-induced rearrangement of the Hofstadter butterfly

The spectrum of 2D electrons subjected to a weak 2D potential and a perpendicular magnetic field is composed of Landau bands with a fractal internal pattern of subbands and minigaps referred to as Hofstadter's butterfly. The Hall conductance may serve as a spectroscopic tool as each filled subband contributes a specific quantized value. Advances in sample fabrication now finally offer access to the regime away from the limiting case of a very weak potential. Complex behavior of the Hall conductance is observed and assigned to Landau band-coupling-induced rearrangements within the butterfly.