Magnetic von Neumann lattice for two-dimensional electrons in a magnetic field

Ishikawa, Kenzo; Maeda, Nobuhiro; Tadaki, K.

doi:10.1103/physrevb.51.5048

Cited by 12 publications

(12 citation statements)

References 9 publications

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“…In fact, the potential energy term becomes a lattice kinetic term if a suitable periodic potential is used for V(x), as shown in Ref. 3. Eigenstates are then extended.…”

Section: A Hamiltonianmentioning

confidence: 99%

“…We confirmed this property of wave functions by solving the equation numerically. 3 Wave functions of energy EϪE l 0 have spatial extensions of a few magnetic lengths. Many short-range impurity problems are studied as easily as an impurity problem if the impurities are dilute and random.…”

Section: ͑22͒mentioning

confidence: 99%

“…1,2 Namely, it was shown that the Hall conductance xy is a topological invariant of a mapping from the momentum space to a space defined by the propagator, and the quantization of xy as (e 2 /h)N at the plateau is thus exact in systems of interactions and disorders. One-particle properties in systems of short-range impurities, a boundary potential, and a periodic array of potentials have been obtained, 3 also based on the same representation. Localization is shown easily in our method.…”

Section: Introductionmentioning

confidence: 99%

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Integer quantum Hall effect with realistic boundary condition: Exact quantization and breakdown

1996

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A theory of the integer quantum Hall effect ͑QHE͒ in realistic systems based on a von Neumann lattice is presented. We show that the momentum representation is quite useful and that the quantum Hall regime ͑QHR͒, which is defined by the propagator in the momentum representation, is realized. In the QHR, the Hall conductance is given by a topological invariant of the momentum space and is quantized exactly. The edge states do not modify the value and topological property of xy in the QHR. We next compute the distribution of current based on the effective action and generally find a finite amount of current in the bulk and the edge. Due to the Hall electric field in the bulk, breakdown of the QHE occurs. The critical electric field of the breakdown is proportional to B 3/2 and the proportional constant has no dependence on Landau levels in our theory, in agreement with recent experiments.

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“…In fact, the potential energy term becomes a lattice kinetic term if a suitable periodic potential is used for V(x), as shown in Ref. 3. Eigenstates are then extended.…”

Section: A Hamiltonianmentioning

confidence: 99%

Section: ͑22͒mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integer quantum Hall effect with realistic boundary condition: Exact quantization and breakdown

1996

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“…<i|Ṽ |j> and <m|V |m ′ > have the same spectrum except for the ground state. Let The problem of an electron in a magnetic field interacting with point impurities has been discussed extensively in the literature [10][11][12][13][14][15]. It has been shown that the zeros of the wave function can be adjusted to coincide with the locations of the scatterers if the concentration of the scatterers is low enough.…”

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confidence: 99%

Energy spectrum for two-dimensional potentials in very high magnetic fields

Gedik

Bayındır

1997

Phys. Rev. B

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A method, analogous to supersymmetry transformation in quantum mechanics, is developed for a particle in the lowest Landau level moving in an arbitrary potential. The method is applied to two-dimensional potentials formed by Dirac ␦ scattering centers. In the periodic case, the problem is solved exactly for rational values of the magnetic flux ͑in units of flux quantum͒ per unit cell. The spectrum is found to be self-similar, resembling the Hofstadter butterfly ͓Phys. Rev. B 14, 2239 ͑1976͔͒. ͓S0163-1829͑97͒06436-9͔In recent years, the energy spectrum of two-dimensional electron systems have attracted great interest, because of the relevance of the problem to the magnetotransport properties 1 and in particular to the quantum Hall effect. 2 It is believed that the physics of the integer quantum Hall effect is governed by the interaction of electrons with a disordered potential, which leads to a localization of the eigenfunctions. Observed conductance steps can be explained by a sequence of localization-delocalization transitions. Quantization of the Hall conductance due to periodic potentials has also been studied. 3,4 In this case, the presence of steps is explained by the gaps in the energy spectrum.There are two opposite approaches to the problem of an electron moving in a periodic potential: the tight-binding and the nearly-free-electron methods. In the first approach, the magnetic field is introduced via Peierls substitution, where the matrix elements are multiplied by exp͓(iq/បc) ͐A•dl͔. 5 On the other hand, in the case of nearly free electrons, the Landau-level structure is essential, and the lattice potential is introduced via intra-and inter-Landau-level scattering matrix elements. The duality between the position and the momentum in quantum mechanics leads to similarities between the two methods. In the presence of a magnetic field, the secular equations for the two limits, the tight-binding and the nearlyfree-electron approach, with certain approximations, are identical. 6 The characteristic feature of the problem is that the secular determinant for the limit of infinite crystals can be reduced to a finite determinant, when the magnetic flux per unit cell is a rational number in units of the flux quantum. The tight-binding case was studied by Azbel' 7 and Hofstadter, 8 who showed that the system has a complicated self-similar spectrum.In this work, an approach is developed for a particle in the lowest Landau level moving in an arbitrary potential. The method is used to obtain the energy spectrum in the presence of Dirac ␦ potentials. The difficulty with the Dirac ␦ potential is that, even if the inter-Landau-level couplings are small and can be neglected, there is a strong intra-Landau-level mixing. The formalism developed in this study leads to an eigenvalue problem where the coupling between distant sites become negligibly small. Moreover, since the problem is formulated in real space, the distribution of Dirac ␦ potentials can be arbitrary. In spite of the sharpness of the potential, the assumption t...

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“…, r N ) is a function of N vector variables, so that one needs a suitable set of basis functions in a 2N -dimensional space. As in the literature there was an extensive discussion on the availability and completeness of different sets of functions in the considered problem [44][45][46][47][48][49][50][51][52], I briefly discuss this point here. The basis set which I will use in this work consists of the eigenfunctions of the kinetic energy operator (15),…”

Section: Trial Solutions Of the Many-body Schrödinger Equationmentioning

confidence: 99%

A new approach to the ground state of quantum Hall systems. Basic principles

Mikhaǐlov

2001

Physica B: Condensed Matter

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I present a new approach to the many-body ground state of quantum-Hall systems. The method describes the behavior of a two-dimensional electron system at all Landau-level filling factors ν, continuously as a function of magnetic field, and enables one to analytically calculate any physical value, characterizing the system, at all ν. New proposed trial many-body wave functions have a clear physical meaning and a very large variational freedom, which opens up wide possibilities to search for the ground state of the system at all magnetic fields.

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Magnetic von Neumann lattice for two-dimensional electrons in a magnetic field

Cited by 12 publications

References 9 publications

Integer quantum Hall effect with realistic boundary condition: Exact quantization and breakdown

Integer quantum Hall effect with realistic boundary condition: Exact quantization and breakdown

Energy spectrum for two-dimensional potentials in very high magnetic fields

A new approach to the ground state of quantum Hall systems. Basic principles

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