Condensation of states at the Landau levels and disorder-independent transport in a random system

I consider an electron on a square lattice, formed by random strenght point potentials, in the presence of a magnetic field, orthogonal to the lattice. When the number of magnetic flux quanta per unit cell is less than one, then an infinite density of extended disorder independent states condenses at each Landau level and provides the possibility for resistivity minima in a disordered system.     

Exact parent Hamiltonian for the quantum Hall states in a lattice

We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.     

Redistributing Chern numbers of Landau subbands: tight-binding electrons under staggered modulated magnetic fields

We investigate the magnetotransport properties of electrons on a square lattice under a magnetic field with the alternate flux strength phi+/-Deltaphi in neighboring plaquettes. A new peculiar behavior of the Hall conductance has been found and is robust against weak disorder: if phi=(p/2N)2pi (p and 2N are coprime integers) is fixed, the Chern numbers of Landau subbands will be redistributed between neighboring pairs and hence the total quantized Hall conductance exhibits a direct transition by +/-Ne2/h at critical fillings when Deltaphi is increased from 0 up to a critical value Deltaphi_{c}. This effect can be an experimental probe of the staggered-flux phase.     

Interband resonant tunneling in superconductor heterostructures in a quantizing magnetic field

The resonant tunneling of electrons through quasistationary levels in the valence band of a quantum well in double-barrier structures based on III–V materials with type-II heterojunctions is considered in a quantizing magnetic field directed perpendicularly to the interfaces. The transmission coefficients of the tunnel structure for transitions from states corresponding to different Landau levels are calculated using the Kane model. It is shown that transitions with a unit change in the Landau level index n as a result of mixing of the wave functions of states with opposite spin orientations are possible on the interfaces due to spin-orbit coupling. The probability of such transitions can be comparable to the probability of transitions without a change in the Landau level index for InAs/AlGaSb/GaSb resonant-tunneling structures. Fiz. Tverd. Tela (St. Petersburg) 40, 2121–2126 (November 1998)     

Exact broken-symmetry states and Hartree–Fock solutions for quantum dots at high magnetic fields

Wigner molecules formed at high magnetic fields in circular and elliptic quantum dots are studied by exact diagonalization (ED) and unrestricted Hartree–Fock (UHF) methods with multicenter basis of displaced lowest Landau level wave functions. The broken symmetry states with semi-classical charge density constructed from superpositions of the ED solutions are compared to the UHF results. UHF overlooks the dependence of the few-electron wave functions on the actual relative positions of electrons localized in different charge puddles and partially compensates for this neglect by an exaggerated separation of charge islands which are more strongly localized than in the exact broken-symmetry states.     

Hofstadter spectrum in electric and magnetic fields

The problem of Bloch electrons in two dimensions subjected to magnetic and intense electric fields is investigated. Magnetic translations, electric evolution, and energy translation operators are used to specify the solutions of the Schrödinger equation. For rational values of the magnetic flux quanta per unit cell and commensurate orientations of the electric field relative to the original lattice, an extended superlattice can be defined and a complete set of mutually commuting space-time symmetry operators is obtained. Dynamics of the system is governed by a finite difference equation that exactly includes the effects of: an arbitrary periodic potential, an electric field orientated in a commensurable direction of the lattice, and coupling between Landau levels. A weak periodic potential broadens each Landau level in a series of minibands, separated by the corresponding minigaps. The addition of the electric field induces a series of avoided and exact crossing of the quasienergies, for sufficiently strong electric field the spectrum evolves into equally spaced discreet levels, in this “magnetic Stark ladder” the energy separation is an integer multiple of hE/aB, with a the lattice parameter.     

Magnetic-Flux-Induced Tunable Completely Flat Band and Topological Nearly Flat Band in Square Kagome Lattice

The flat bands in a square kagome lattice containing square and triangle plaquettes, are respectively investigated, and the origin of the doubly degenerated completely flat bands and the corresponding compact localized states are elucidated. It is found that the introduction of external magnetic flux enriches the modulation parameters, making the system present many interesting results. In the case that the magnetic flux penetrates through each square plaquette, the tunable completely flat band has one more tunable parameter. And when the external magnetic flux penetrates through each triangle plaquette, excepting a completely flat band, the band dispersions of the system present the topological nearly flat band, which is very useful to realize the interesting fractional quantum Hall physics. The average density of states is also calculated to corroborate the completely FB generated by the highly localized eigenstates. Furthermore, the implementation of the square kagome lattice system based on the current photonic waveguide network technology is demonstrated. The scheme opens up a way to generate the tunable completely flat band and topological nearly flat band in square kagome lattice under multi-parameter variable conditions.     

Fermion-pairing on a square lattice in extreme magnetic fields

We consider the Cooper-problem on a two-dimensional, square lattice with a uniform, perpendicular magnetic field. Only rational flux fractions are considered. An extended (real-space) Hubbard model including nearest and next nearest neighbor interactions is transformed to “k-space”, or more precisely, to the space of eigenfunctions of Harper’s equation, which constitute basis functions of the magnetic translation group for the lattice. A BCS-like truncation of the interaction term is performed. Expanding the interactions in the basis functions of the irreducible representations of the point group C_4ν of the square lattice simplify calculations. The numerical results indicate enhanced binding compared to zero magnetic field, and thus re-entrant superconducting pairing at extreme magnetic fields, well beyond the point where the usual semi-classical treatment of the magnetic field breaks down.     

Lattice relaxation of graphene under high magnetic field

We investigate the n = 0 Landau level (LL) in monolayer graphene with high magnetic field. We find that the energy gap is opened in the n = 0 LL by the magnetic-field-dependent lattice relaxation originating from the interactions between the electrons (holes) and longitudinal-deformation-acoustic phonon. Both the linear and square-foot dependence of the energy gap on the magnetic field are obtained depending on the choice of the Debye cut-off wave number for the acoustic phonon. The relations of the Huang-Rhys parameter (lattice relaxation strength) and the transition linewidths with the magnetic field are also discussed. Our results agree with the current experiments on graphene in high magnetic field, and provide an alternative explanation for the experimental measurements.     

Indirect magnetic exchange interaction mediated by bound Landau states in 2DES

The possibility of indirect exchange coupling mediated by Landau electrons bound to magnetic impurities in 2DES is studied here. The importance of the resonance scattering of the Landau electrons with the impurities is emphasized due to its spin selectivity which results in strong spin polarization of the localized Landau states. The bound Landau states act as mediators of the superexchange interaction resulting in an antiferromagnetic interaction between the nuclear spins of the impurities. The coupling constant, between these nuclear spins, J, is presented for the case of a weak scattering limit and found to depend strongly on the ratio of the impurity separation over the magnetic length. Possible applications of these results may include a long-range mechanism for coupling between two nuclear spins to be used as a qubits interaction with a spacing distance of the order of the magnetic length.     

Electronic properties of a Penrose tiling lattice in a magnetic field

The one-electron energy spectrum of a two-dimensional Penrose tiling lattice in a uniform magnetic field is calculated as a function of magnetic fields with a tight-binding Hamiltonian. The calculated results show the following remarkable features characteristic of the Penrose lattice. (1) The density of states in a magnetic field has a central peak with zero width at the zero energy. It is shown that the zero-energy states correspond to the ring states in which the wavefunction has a non-vanishing amplitudes only at the sites on a ring-like region around the origin. (2) The energy levels coalesce into Landau type levels and the boundary states due to the finite size effects based on a fixed boundary condition appear in the gap region between Landau levels. (3) The magnetic field dependence of the energy spectrum has a repeated pattern of self-similarity with the golden mean ratio of two successive periods.     

Landau levels line widths in potential scattering in the presence of a strong magnetic field

Summary The FBA scattering cross-section in the presence of a strong magnetic field diverges at the Landau thresholds. Such divergences are eliminated by the introduction of a modified density of states, accounting for the finite Landau states lifetime of the electrons in a magnetized plasma.     

Self-assembled nanocrystalline CdSe thin films

The topic of this contribution is the investigation of quantum states and quantum Hall effect in electron gas subjected to a periodic potential of the lateral lattice. The potential is formed by triangular quantum antidots located on the sites of the square lattice. In such a system the inversion center and the four-fold rotation symmetry are absent. The topological invariants which characterize different magnetic subbands and their Hall conductances are calculated. It is shown that the details of the antidot geometry are crucial for the Hall conductance quantization rule. The critical values of lattice parameters defining the shape of triangular antidots at which the Hall conductance is changed drastically are determined. We demonstrate that the quantum states and Hall conductance quantization law for the triangular antidot lattice differ from the case of the square lattice with cylindrical antidots. As an example, the Hall conductances of magnetic subbands for different antidot geometries are calculated for the case when the number of magnetic flux quanta per unit cell is equal to three.     

Point impurities remove degeneracy of the Landau levels in a two-dimensional electron gas

The density of states of a two-dimensional electron gas in a magnetic field has been studied taking into account the scattering on point impurities. It is demonstrated that allowance for the electron-impurity interaction completely removes degeneracy of the Landau levels even for a small volume density of these point defects. The density of states is calculated in a self-consistent approximation taking into account all diagrams without intersections of the impurity lines. The electron density of states ρ is determined by the contribution from a cut of the one-particle Green’s function rather than from a pole. In a wide range of the electron energies ω (measured from each Landau level), the value of ρ(ω) is inversely proportional to the energy |ω| and proportional to the impurity concentration. The obtained results are applicable to various two-dimensional electron systems such as inversion layers, heterostructures, and electrons on the surface of liquid helium.     

Evidence for skyrmion crystallization from NMR relaxation experiments

A resistively detected NMR technique was used to probe the two-dimensional electron gas in a GaAs/AlGaAs quantum well. The spin-lattice relaxation rate (1/T(1)) was extracted at near complete filling of the first Landau level by electrons. The nuclear spin of (75)As is found to relax much more efficiently with T --> 0 and when a well developed quantum Hall state with R(xx) approximately 0 occurs. The data show a remarkable correlation between the nuclear spin relaxation and localization. This suggests that the magnetic ground state near complete filling of the first Landau level may contain a lattice of topological spin texture, i.e., a Skyrmion crystal.     

“Soft” modes of the excitation spectrum constructed on perturbations of the Abrikosov lattice with a single flux quantum in the unit cell

We have analyzed the spectrum of gapless excitations emerging upon the perturbation of the Abrikosov lattice with a single flux quantum in the unit cell. Superconductors with Ginzburg–Landau parameter κ close to unity are of special interest. We have determined the spectrum of gapless excitations close to zeroth shear modes for an arbitrary angle ? between the unit cell vectors. Analysis of the excitation spectra of triangular and square lattices with a single flux quantum in the unit cell has shown that solutions with a number of flux quanta greater than one exist at least in the range of parameters κ close to unity (κ > 1) and give smaller values of the free energy as compared to its values for a triangular lattice with a single flux quantum. For small values of momentum k (in the k ² approximation), the excitation spectrum of the “transverse” mode in the triangular lattice is independent of the direction of the momentum lying in the plane perpendicular to the magnetic field. For the square lattice (? = π/2), the transverse mode is anisotropic in the k ² approximation also.     

Impurity states of electrons in quantum dots in external magnetic fields

The influence of isolated impurity atoms on the electron energy spectrum in a parabolic quantum dot in quantizing magnetic field is studied. The impurity potential is approximated by a Gaussian separable operator which allows one to obtain the exact solution of the problem. We demonstrate that in the electron energy spectrum there is a set of local levels which are split from the Landau zone boundaries in the upward or downward direction depending on the impurity type. We have calculated the local level positions, the wave functions of electrons in bound states, and the residues of the electron scattering amplitudes by impurity atoms at the poles.     

Light scattering by a 2D electron system with spin-orbit interaction in a strong magnetic field

We theoretically investigate light scattering by electrons of a 2D system in a strong magnetic field perpendicular to the plane of the system (the filling factor is smaller than two, i.e., only the states of the zeroth Landau level are filled). Analysis is carried out in the resonance approximation, in which the frequencies of the incident and scattered light are close to the effective separation between the conduction band and one of the branches of the valence band of the semiconductor. Spin splitting of the Landau level in the conduction band associated with Zeeman and the spin-orbit interactions is taken into account. The dynamic screening effects are considered in the random phase approximation (RPA).