Higher Landau Levels Research Articles

Direct observation of a magnetic-field-induced Wigner crystal.

Wigner predicted that when the Coulomb interactions between electrons become much stronger than their kinetic energy, electrons crystallize into a closely packed lattice1. A variety of two-dimensional systems have shown evidence for Wigner crystals2-11 (WCs). However, a spontaneously formed classical or quantum WC has never been directly visualized. Neither the identification of the WC symmetry nor direct investigation of its melting has been accomplished. Here we use high-resolution scanning tunnelling microscopy measurements to directly image a magnetic-field-induced electron WC in Bernal-stacked bilayer graphene and examine its structural properties as a function of electron density, magnetic field and temperature. At high fields and the lowest temperature, we observe a triangular lattice electron WC in the lowest Landau level. The WC possesses the expected lattice constant and is robust between filling factor ν ≈ 0.13 and ν ≈ 0.38 except near fillings where it competes with fractional quantum Hall states. Increasing the density or temperature results in the melting of the WC into a liquid phase that is isotropic but has a modulated structure characterized by the Bragg wavevector of the WC. At low magnetic fields, the WC unexpectedly transitions into an anisotropic stripe phase, which has been commonly anticipated to form in higher Landau levels. Analysis of individual lattice sites shows signatures that may be related to the quantum zero-point motion of electrons in the WC lattice.

Cascade of Multielectron Bubble Phases in Monolayer Graphene at High Landau Level Filling

Cascade of Multielectron Bubble Phases in Monolayer Graphene at High Landau Level Filling

Entanglement entropy and hyperuniformity of Ginibre and Weyl–Heisenberg ensembles

We show that, for a class of planar determinantal point processes (DPP) {mathcal {X}}, the growth of the entanglement entropy S({mathcal {X}}(Omega )) of {mathcal {X}} on a compact region Omega subset {mathbb {R}}^{2d}, is related to the variance {mathbb {V}}left( {mathcal {X}}(Omega )right) as follows: VX(Ω)≲SX(Ω)≲VX(Ω).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {V}}\\left( {\\mathcal {X}}(\\Omega )\\right) \\lesssim S\\left( \\mathcal {X(} \\Omega \\mathcal {)}\\right) \\lesssim {\\mathbb {V}}\\left( {\\mathcal {X}}(\\Omega )\\right) . \\end{aligned}$$\\end{document}Therefore, such DPPs satisfy an area lawSleft( {mathcal {X}}_{g} mathcal {(}Omega mathcal {)}right) lesssim left| partial {Omega } right| , where partial {Omega } is the boundary of Omega if they are of Class I hyperuniformity ({mathbb {V}}left( {mathcal {X}} (Omega )right) lesssim left| partial {Omega }right| ), while the area law is violated if they are of Class II hyperuniformity (as Lrightarrow infty , {mathbb {V}}left( {mathcal {X}} (LOmega )right) sim C_{Omega }L^{d-1}log L). As a result, the entanglement entropy of Weyl–Heisenberg ensembles (a family of DPPs containing the Ginibre ensemble and Ginibre-type ensembles in higher Landau levels), satisfies an area law, as a consequence of its hyperuniformity.

Analysis of Landau level contribution to pseudo chiral magnetic effect in Weyl–Dirac matter

This study is motivated by the recent observed realization of Pseudo Chiral Magnetic Effect (PCME) into the continuum limit of the low-energy tight binding description of graphene. We here proposed the quantum description of PCME by investigating the Landau level structure of the system. The different physical quantities like current, chiral currents, number densities and chiral condensate are discussed in this context with this Landau level description of the system. Further, the effect of different physical conditions to the system is studied by analyzing the behavior of Landau levels under these conditions. This provides a microscopic understanding of the phenomenon. It has been shown that largest contribution to the PCME currents and number densities comes from the spin polarized lowest Landau level which corresponds to the system characteristics at low-temperature and high magnetic field. However, the contribution to the PCME current from the higher Landau levels is just opposite to that from the lowest Landau level. We have further established general expressions for all these physical quantities regarding to any higher-order Landau level.

Local Maxima of White Noise Spectrograms and Gaussian Entire Functions

We confirm Flandrin’s prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions), a consequence of the conjectured double honeycomb mean model for the network of zeros and local maxima, where the area of local maxima centered hexagons is three times larger than the area of zero centered hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their expected density of zeros is 1, have an expected density of 5/3 critical points, among those 1/3 are local maxima, and 4/3 saddle points, and compute the distributions of ordinate values (heights) for spectrogram local extrema. This is done by first writing the spectrograms in terms of Gaussian Entire Functions (GEFs). The extrema are considered under the translation invariant derivative of the Fock space (which in this case coincides with the Chern connection from complex differential geometry). We also observe that the critical points of a GEF are precisely the zeros of a Gaussian random function in the first higher Landau level. We discuss natural extensions of these Gaussian random functions: Gaussian Weyl–Heisenberg functions and Gaussian bi-entire functions. The paper also reviews recent results on the applications of white noise spectrograms, connections between several developments, and is partially intended as a pedestrian introduction to the topic.

Two-dimensional finite quantum Hall clusters of electrons with anisotropic features

Low-dimensional nano and two-dimensional materials are of great interest to many disciplines and may have a lot of applications in fields such as electronics, optoelectronics, and photonics. One can create quantum Hall phases by applying a strong magnetic field perpendicular to a two-dimensional electron system. One characterizes the nature of the system by looking at magneto-transport data. There have been a few quantum phases seen in past experiments on GaAs/AlGaAs heterostructures that manifest anisotropic magnetoresistance, typically, in high Landau levels. In this work, we model the source of anisotropy as originating from an internal anisotropic interaction between electrons. We use this framework to study the possible anisotropic behavior of finite clusters of electrons at filling factor 1/6 of the lowest Landau level.

Machine learning study for disorder effect at a half-filled high Landau level

The robustness against disorder scattering is crucial for experimentally observing the predicted charge density waves (CDWs) in a fractional quantum Hall (FQH) system with partially-filled topmost Landau level (LL). Here, we applied two types of machine learning (ML) methods to study the influence of the disorder on an example system with half-filled N = 2 LL. Through the unsupervised principal component analysis (PCA) method, we recognize that a CDW stripe phase is represented by two principal components, which alternatively dominate in weak scattering case and coherently collapse at strong scattering. A combination of these two PCA components enables us to fully describe the evolution of the stripe phase and to determine its critical transition point towards a random disorder phase. A practice with the supervised neural network (NN) method also provides us with the numerically same boundary for the two separated phases. These ML approaches have proved to be effective tools for our system as their results are well compatible with previous numerical works. Moreover, the nonnecessity to the explicit knowledge of the states extends their potential to study other unfamiliar disordered systems.

Orbital mixing in few-layer graphene and non-Abelian Berry phase

In a magnetic field few-layer graphene supports, at the lowest Landau level, a multiplet of zero-mode levels nearly degenerate in orbitals as well as in spins and valleys. Those pseudo-zero-mode (PZM) levels are generally sensitive to interactions and external perturbations, and have a crossing among themselves or with other higher Landau levels when an external field is swept over a certain range. A close study is made of how such PZM levels evolve when they are gradually brought from empty to filled levels under many-body interactions. It is pointed out that the level spectra generally avoid a crossing via orbital level mixing and that orbital mixing is governed by a non-Abelian Berry phase that derives from an approximate degeneracy and interactions. A look is also taken into evolution/crossing of many-body ground states with increasing external bias in bilayer graphene.

Stability of even-denominator fractional quantum Hall states in systems with strong Landau-level mixing

Mixing of Landau levels has been understood to be essential in governing the nature of the ground state for the even-denominator fractional quantum Hall effect. The incompressibility of the ground state at filling factor $5/2$ in the strong Landau-level-mixed systems, such as the ZnO quantum well, is not always stable. Here we present an approach to generally deal with this kind of system and satisfactorily explain the recent experiments [Falson et al., Sci. Adv. 4, eaat8742 (2018)] by implementing the screening plus the thickness effect. Further, the phase diagrams of the incompressibility of the ground state indicate that the phase transitions can be explicitly extracted by observing the lowest gap of the collective modes when the magnetic field and the width of the quantum well are tuned. We also predict the incompressibility of the two-dimensional electron gas in higher Landau levels in another strong Landau-level-mixed system, viz., the black phosphorene, by considering the screening effect where the relevant even-denominator fractional quantum Hall effects can possibly be observed.

Melting phase diagram of bubble phases in high Landau levels

A low-disorder, two-dimensional electron system (2DES) subjected to a large perpendicular magnetic field and cooled to very low temperatures provides a rich platform for studies of many-body quantum phases. The magnetic field quenches the electrons' kinetic energy and quantizes the energy into a set of Landau levels, allowing the Coulomb interaction to dominate. In excited Landau levels, the fine interplay between short- and long-range interactions stabilizes bubble phases, Wigner crystals with more than one electron per unit cell. Here we present the screening properties of bubble phases, probed via a simple capacitance technique where the 2DES is placed between a top and a bottom gate and the electric field penetrating through the 2DES is measured. The bubbles formed at very low temperatures screen the electric field poorly as they are pinned by the residual disorder potential, allowing a large electric field to reach the top gate. As the temperature is increased, the penetrating electric field decreases and, surprisingly, exhibits a pronounced minimum at a temperature that appears to coincide with the melting temperature of the bubble phase. We deduce a quantitative phase diagram for the transition from bubble to liquid phases for Landau level filling factors $4\leq\nu\leq5$.

Limits of Applicability of the Composite Fermion Model.

The popular model of composite fermions, proposed in order to rationalize FQHE, were insufficient in view of recent experimental observations in graphene monolayer and bilayer, in higher Landau levels in GaAs and in so-called enigmatic FQHE states in the lowest Landau level of GaAs. The specific FQHE hierarchy in double Hall systems of GaAs 2DES and graphene also cannot be explained in the framework of composite fermions. We identify the limits of the usability of the composite fermion model by means of topological methods, which elucidate the phenomenological assumptions in composite fermion structure and admit further development of FQHE understanding. We demonstrate how to generalize these ideas in order to explain experimentally observed FQHE phenomena, going beyond the explanation ability of the conventional composite fermion model.

Magnetotransport patterns of collective localization near ν=1 in a high-mobility two-dimensional electron gas

We report complex magnetotransport patterns of the $\nu=1$ integer quantum Hall state in a GaAs/AlGaAs sample from the newest generation with a record high electron mobility. The reentrant integer quantum Hall effect in the flanks of the $\nu=1$ plateau indicates the formation of the integer quantum Hall Wigner solid, a collective insulator. Moreover, at a fixed filling factor, the longitudinal resistance versus temperature in the region of the integer quantum Hall Wigner solid exhibits a sharp peak. Such sharp peaks in the longitudinal resistance versus temperature so far were only detected for bubble phases forming in high Landau levels but were absent in the region of the Anderson insulator. We suggest that in samples of sufficiently low disorder sharp peaks in the longitudinal resistance versus temperature traces are universal transport signatures of all isotropic electron solids that form in the flanks of integer quantum Hall plateaus. We discuss possible origins of these sharp resistance peaks and we draw a stability diagram for the insulating phases in the $\nu$-$T$ phase space.

Thermoelectric properties of the (an-)isotropic QGP in magnetic fields

The Seebeck effect and the Nernst effect, which reflect the appearance of electric fields along x-axis and along y-axis (E_{x} and E_{y}), respectively, induced by the thermal gradient along x-axis, are studied in the QGP at an external magnetic field along z-axis. We calculate the associated Seebeck coefficient (S_{xx}) and Nernst signal (N) using the relativistic Boltzmann equation under the relaxation time approximation. In an isotropic QGP, the influences of magnetic field (B) and quark chemical potential (mu _{q}) on these thermoelectric transport coefficients are investigated. In the presence (absence) of weak magnetic field, we find S_{xx} for a fixed mu _{q} is negative (positive) in sign, indicating that the dominant carriers for converting heat gradient to electric field are negatively (positively) charged quarks. The absolute value of S_{xx} decreases with increasing temperature. Unlike S_{xx}, the sign of N is independent of charge carrier type, and its thermal behavior displays a peak structure. In the presence of strong magnetic field, due to the Landau quantization of transverse motion of (anti-)quarks perpendicular to magnetic field, only the longitudinal Seebeck coefficient (S_{zz}) exists. Our results show that the value of S_{zz} at a fixed mu _{q} in the lowest Landau level (LLL) approximation always remains positive. Within the effect of high Landau levels, S_{zz} exhibits a thermal structure similar to that in the LLL approximation. As the Landau level increases further, S_{zz} decreases and even its sign changes from positive to negative. The computations of these thermoelectric transport coefficients are also extended to a medium with momentum-anisotropy induced by initial spatial expansion as well as strong magnetic field.

Size scaling of the exchange interaction in the quantum Hall effect regime

We use the self-consistent Hartree-Fock approximation for numerically addressing the integer quantum Hall effect (IQHE) regime in terms of many-body physics at higher Landau levels (LLs). We investigate the dependence of many-particle interactions on the lateral size of the electron system. We use the exchange enhancement of the $g$-factor for spin-polarized Landau levels as an indicator for the strength of the exchange interaction. The driving force for the $g$-factor enhancement is a Hund's rule behavior for the occupation of spin-split Landau levels that lowers the many-particle ground state energy by arranging as many spins in parallel as possible. By increasing the total number of electrons and total number of available states per LL, it can therefore be expected that the exchange-enhanced spin gap should increase as well. In contrast to the dependence on the magnetic field, an increase of the total number of states by simply increasing the system size at constant magnetic field shows a clear saturation behavior above a lateral system size of 1000 nm. The importance of this result is underlined by an extended introduction, which demonstrates the permanent dominance of many-body interactions in all transport regimes of the IQHE. A modeling of IQHE systems therefore has to include many-body interactions, and our results open a pathway towards many-body modeling of quantum Hall systems of macroscopic size.

Hidden Quantum Hall Stripes in Al_{x}Ga_{1-x}As/Al_{0.24}Ga_{0.76}As Quantum Wells.

We report on transport signatures of hidden quantum Hall stripe (hQHS) phases in high (N>2) half-filled Landau levels of Al_{x}Ga_{1-x}As/Al_{0.24}Ga_{0.76}As quantum wells with varying Al mole fraction x<10^{-3}. Residing between the conventional stripe phases (lower N) and the isotropic liquid phases (higher N), where resistivity decreases as 1/N, these hQHS phases exhibit isotropic and N-independent resistivity. Using the experimental phase diagram, we establish that the stripe phases are more robust than theoretically predicted, calling for improved theoretical treatment. We also show that, unlike conventional stripe phases, the hQHS phases do not occur in ultrahigh mobility GaAs quantum wells but are likely to be found in other systems.

Microscopic details of stripes and bubbles in the quantum Hall regime

We use a fully self-consistent laterally resolved Hartree-Fock approximation for numerically addressing the electron configurations at higher Landau levels in the quantum Hall regime for near-macroscopic sample sizes. Our results give microscopic details of stripe- and bubble-like charge density modulations and show how these emerge depending on the filling factor. We find that there exists a region at the boundaries of the stripes and bubbles with a density modulation that corresponds to a filling factor around half filling. The microscopic details of these boundary regions determine the geometrical boundary conditions for aligning the charge density modulation either as stripes or bubbles. Transport is modelled using a non-equilibrium network model giving a pronounced anisotropy in direction of the injected current in the stripe regime close to half filling. We obtain a stripe period of 2.9 cyclotron radii. Our results indicate the dominance of many particle physics in the integer quantum Hall regime and provide an intuitive understanding of its consequences in strong magnetic fields.

Entanglement entropy for integer quantum Hall effect in two and higher dimensions

We analyze the entanglement entropy, in real space, for the higher dimensional integer quantum Hall effect on ${\mathbb {CP}}^k$ (any even dimension) for abelian and nonabelian magnetic background fields. In the case of $\nu=1$ we perform a semiclassical calculation which gives the entropy as proportional to the phase-space area. This exhibits a certain universality in the sense that the proportionality constant is the same for any dimension and for any background, abelian or nonabelian. We also point out some distinct features in the profiles of the eigenfunctions of the two-point correlator that underline the difference in the value of entropies between $\nu=1$ and higher Landau levels.

Holomorphic quantum Hall states in higher Landau levels

Eigenstates of the planar magnetic Laplacian with a homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems; in particular, we represent effective Hamiltonians and particle densities in higher Landau levels by using corresponding quantities in the lowest Landau level.

Anomalous Nematic States in High Half-Filled Landau Levels

It is well established that the ground states of a two-dimensional electron gas with half-filled high (N≥2) Landau levels are compressible charge-ordered states, known as quantum Hall stripe (QHS) phases. The generic features of QHSs are a maximum (minimum) in a longitudinal resistance R_{xx} (R_{yy}) and a nonquantized Hall resistance R_{H}. Here, we report on emergent minima (maxima) in R_{xx} (R_{yy}) and plateaulike features in R_{H} in half-filled N≥3 Landau levels. Remarkably, these unexpected features develop at temperatures considerably lower than the onset temperature of QHSs, suggestive of a new ground state.

On elastic scattering amplitude of planar charged fermions in a constant magnetic field

The elastic scattering amplitude (ESA) is obtained in the one-loop approximation of the (2[Formula: see text]+[Formula: see text]1)-dimensional quantum electrodynamics (QED[Formula: see text]) for planar charged fermions in an external constant magnetic field. We obtain the elastic scattering amplitude in the corresponding massive theory and then discuss and calculate the massless limit of ESA. The imaginary part of ESA are related to the total probability of photons emission by charged fermions in the considered magnetic field. Simple analytical formulas are obtained when planar charged fermions in initial states occupy high excited Landau levels.