Lowest Landau Band Research Articles

Impurity and dispersion effects on the linear magnetoresistance in the quantum limit

Magnetoresistance, that is, the change of the resistance with the magnetic field, is usually a quadratic function of the field strength. A linear magnetoresistance usually reveals extraordinary properties of a system. In the quantum limit where only the lowest Landau band is occupied, a quantum linear magnetoresistance was believed to be the signature of the Weyl fermions with 3D linear dispersion. Here, we comparatively investigate the quantum-limit magnetoresistance of systems with different band dispersions as well as different types of impurities. We find that the magnetoresistance can also be linear for the quadratic energy dispersion. We show that both longitudinal and transverse magnetoresistance can be linear if long-range-Gaussian-type impurities dominate, but Coulomb-type impurities can only induce linear transverse magnetoresistance. Moreover, we find a negative longitudinal magnetoresistance in massless Dirac fermions, regardless of the impurity type, as a result of the combined effect of the linear dispersion and the scattering mechanism. Our findings well explain some of the linear magnetoresistance observed in the experiments and provide insights to the understanding of quantum-limit magnetoresistance.

Edge-state critical behavior of the integer quantum Hall transition

The integer quantum Hall effect features a paradigmatic quantum phase transition. Despite decades of work, experimental, numerical, and analytical studies have yet to agree on a unified understanding of the critical behavior. Based on a numerical Green function approach, we consider the quantum Hall transition in a microscopic model of non-interacting disordered electrons on a simple square lattice. In a strip geometry, topologically induced edge states extend along the system rim and undergo localization-delocalization transitions as function of energy. We investigate the boundary critical behavior in the lowest Landau band and compare it with a recent tight-binding approach to the bulk critical behavior [Phys. Rev. B 99, 121301(R) (2019)] as well as other recent studies of the quantum Hall transition with both open and periodic boundary conditions.

Anti-levitation in integer quantum Hall systems

The evolution of extended states of two-dimensional electron gas with white-noise randomness and field is numerically investigated by using the Anderson model on square lattices. Focusing on the lowest Landau band we establish an anti-levitation scenario of the extended states: As either the disorder strength $W$ increases or the magnetic field strength $B$ decreases, the energies of the extended states move below the Landau energies pertaining to a clean system. Moreover, for strong enough disorder, there is a disorder-dependent critical magnetic field ${B}_{c}(W)$ below which there are no extended states at all. A general phase diagram in the $W\text{\ensuremath{-}}1/B$ plane is suggested with a line separating domains of localized and delocalized states.

One-Dimensional Models for Atoms in Strong Magnetic Fields, II: Anti-Symmetry in the Landau Levels

Electrons in strong magnetic fields can be described by one-dimensional models in which the Coulomb potential and interactions are replaced by regularizations associated with the lowest Landau band. For a large class of models of these type, we show that the maximum number of electrons that can be bound is less than a Z + Z f(Z). The function f(Z) represents a small non-linear growth which is quadratic in log Z when the magnetic field strength grows polynomially with the nuclear charge Z. In contrast to earlier work, the models considered here include those arising from realistic cases in which the full trial wave function for N-electrons is the product of an N-electron trial function in one-dimension and an antisymmetric product of states in the lowest Landau level.

Gradient corrections for semiclassical theories of atoms in strong magnetic fields

This paper is divided into two parts. In the first one the von Weizsäcker term is introduced to the magnetic Thomas–Fermi theory and the resulting MTFW functional is mathematically analyzed. In particular, it is shown that the von Weizsäcker term produces the Scott correction up to magnetic fields of order B≪Z2, in accordance with a result of Ivrii on the quantum mechanical ground state energy. The second part is dedicated to gradient corrections for semiclassical theories of atoms restricted to electrons in the lowest Landau band. We consider modifications of the Thomas–Fermi theory for strong magnetic fields (STF), i.e., for B≪Z3. The main modification consists in replacing the integration over the variables perpendicular to the field by an expansion in angular momentum eigenfunctions in the lowest Landau band. This leads to a functional (DSTF) depending on a sequence of one-dimensional densities. For a one-dimensional Fermi gas the analogue of a Weizsäcker correction has a negative sign and we discuss the corresponding modification of the DSTF functional.

Bounds on One-Dimensional Exchange Energies with Application to Lowest Landau Band Quantum Mechanics

By means of a generalization of the Fefferman–de la Llave decomposition we derive a general lower bound on the interaction energy of one-dimensional quantum systems. We apply this result to a specific class of lowest Landau band wave functions.

Novel Scaling Relation of the Energy Spacing Distribution in Quantum-Hall Systems

The shape analysis of the energy spacing distribution $P(s)$ obtained from numerical simulation of two dimensional disordered electron systems subject to strong magnetic fields is performed. In the present work we reanalyze the data obtained in a previous publication. Special moments of the $P(s)$ function related to R\'enyi-entropy differences show a novel scale invariant relation that is attributed to the presence of one-parameter scaling. This relation seems to show both deviations and universality depending on which Landau-band is considered and whether the disorder is correlated or uncorrelated. Furthermore, our analysis shows the existence of an huge, however, irrelevant length scale in the case of the second lowest Landau-band and no disorder correlations that completely disappears with the introduction of disorder correlations on the range of one magnetic length.

Fractional quantum Hall effects as an example of fractal geometry in nature

A prescription is provided for constructing the Hall curve including both integral (I)- and fractional (F)-quantum Hall effects (QHE) that is based upon the iterative application of particular transformations simultaneously to the Hall resistance (Rxy) and magnetic field (B) axes of a template constructed from the elementary (integral quantum) Hall curve to filling factor v = 1. The construction shows that scaled copies of the elementary Hall curve reappear in various parts of the constructed curve upon increasing the magnification, resulting in FQHE sequences in higher Landau bands, and novel FQHE sequences between main sequence FQHE’s in the lowest Landau band. The self similarity observed in the constructed Hall curve helps to draw a connection between FQHE’s and the classical problem of an electronin-a-periodic-potential-subjected-to-a-magnetic-field (‘Hofstadter’s butterfly’), and suggests that fractional quantum Hall effects constitute another manifestation of fractal geometry in nature—one that might also be viewed as a signature of transport in a Wigner crystal.

Scaling Behavior of Level Statistics in Quantum Hall Regime

The scaling property of level statistics in the quantum Hall regime, i.e. 2D disordered electron systems subject to strong magnetic fields, is analyzed numerically in the light of the random matrix theory. The energy dependences of the effective level repulsion parameter, the two level correlation, the GUE-GOE crossover parameter, and the rigidity (or $\Delta_3$-statistics) of the level distributions are investigated for different system sizes by unfolding the original data and by dividing the unfolded spectrum into small regions. It is shown that the critical exponent of the localization length as a function of energy can be determined through the energy dependence of the level statistics. The analyses are carried out not only for the lowest Landau band (LB) but also for the second lowest LB. Furthermore the effect of finite range of disordered potential is studied. The short-ranged potential case in the second lowest LB is found to be pathological as in other studies of critical behavior, and it is confirmed that this pathological behavior is improved in the case of disordered potential with finite ranges.

Electron-electron interaction and the persistent current in a quantum ring.

We have studied the effect of electron-electron interaction on the magnetic moment (associated with the persistent current) of electrons in a quantum ring. We have introduced a model where the electron makes a circular motion in a parabolic confinement simulating a quantum ring which is subjected to a perpendicular magnetic field. The electron states in such a ring with and without the Coulomb interaction are then investigated. We have also explored the limits of narrow and wide rings. Our detailed calculations of the energy spectrum where the interelectron interactions are also included indicate that when the electrons occupy the states of only the lowest Landau band, the interaction merely shifts the spectrum to higher energy and its effect on magnetization is insignificant. The result can be understood as entirely due to conservation of the angular momentum. A qualitative estimate for a large system where electrons also occupy higher Landau bands indicates that the Coulomb interaction mixes some of the states resulting in a rapid increase in the magnetization.

Delocalization transition of two-dimensional disordered systems in a strong magnetic field

The delocalization transition in two-dimensional systems and a strong magnetic field is investigated with respect to its dependence on the Landau band indexj and on the type of disorder. The generation of random potentials according to a given correlation functionf and for a chosen correlation lengthd is described. The spectral properties of random eigenvalue sequences are examined as measures for the extension of wavefunctions and indicate a nonuniversal delocalization behaviour in higher Landau bands for short ranged correlated potentials. The critical exponents of the localization length of wavefunctions are determined for rapidly varying potentials in the second lowest Landau band (j=1) and depend on the correlation lengthd of the disorder. This different critical behaviour compared to that in the lowest band is confirmed by calculations for the density-density correlations of wavefunctions at the centers of the Landau levels. Calculations in different geometries also show that the critical systems of delocalized states are conformal invariant in the case of the nonuniversal delocalization transition (d≲l0), whereas such local rescaling properties cannot be expected for slowly varying potentials.

Scaling Behavior of Level Spacing Distribution in the Lowest Landau Band of Two-Dimensional Disordered Electrons

The spacing distributions of unfolded energy levels in the lowest Landau band formed in disordered two-dimensional electron systems are numerically studied for different energy regions. By fitting the distribution function to an appropriate interpolation formula, the level repulsion parameter β is obtained as a function of the unfolded energy x . The functional forms of β( x ) for different system sizes are found to exhibit a certain scaling behavior. From this scaling property, it is possible to estimate the exponent of the localization length as a function of energy, which is found to be 2.4±0.2, consistent with previously reported values. Possible applications to other Anderson transitions are also discussed.

Level Spacing Distribution and Δ3-Statistics of Two Dimensional Disordered Electrons in Strong Magnetic Field

The statistical properties of the energy levels of the lowest Landau band are analyzed. The eigenvalues are obtained numerically for a random matrix model that describes a 2D disordered electron system in a strong magnetic field. In order to avoid the problems related to an energy region dependent average level spacing, the original data are unfolded using the ensemble averaged integrated density of states. For the unfolded data, the level spacing distribution and the Δ 3 -statistics are investigated in order to clarify the level correlations. The statistical properties depend on the energy and the magnitude of the level separation. A continuous change from Poissonian to Gaussian unitary statistics is observed between the edge and the center of the band. For the statistics within a given energy region, a highly non-trivial crossover of symmetries takes place between small and large level separations. Even if the localization length is much larger than the system size the deviations from the Gaussian unita...

Two-dimensional disordered electronic systems in a strong magnetic field.

We study electron states at the center of the lowest Landau band when the impurities form a point set of finite density. The electron wave function is expressed in terms of an entire function of order 2 and finite type that vanishes at all the complex locations z=x+iy of the impurities. Two fields B 1 ≤B 2 are found such that B>B 1 is necessary and B>B 2 sufficient for the existence of regular solutions, which, in the latter case, are given explicitly. Between these two values there exists a critical field B c such that solutions exist if B>B c and do not exist if B<B x

Analysis of Two Dimensional Electronic States in Strong Magnetic Field by Random Matrix Model

Electronic states in a two dimensional random system with a perpendicular strong magnetic field are analyzed by using a model Hamiltonian matrix which has random elements reflecting the characteristics of the random potential such as the correlation length. The random matrix model is suitable for numerical calculations and has advantages compared to preparing the random potential by distributing impurities particularly when discussing correlated potential. As an example, the fractal dimensionality of extended states in the lowest Landau band is considered. It is found that the extended states have multifractal properties. The results are compared with those obtained through the random potential model.

Density of states in a uniform magnetic field and a white noise potential

The author shows that F. Wegner's (1983) calculation of the density of states in the lowest Landau band agrees, in the tail, with the semiclassical result. The complex nature of the classical solutions governing the density of states between Landau bands is discussed and some numerical results are given.