Von Neumann Lattice Research Articles

Optimal bases of Gaussians in a Hilbert space: applications in mathematical signal analysis

Arbitrary square-integrable (normalized) functions can be expanded exactly in terms of the Gaussian basis g( t; A) where A∈ C. Smaller subsets of this highly overcomplete basis can be found, which are also overcomplete, e.g., the von Neumann lattice g( t; A mn ) where A mn are on a lattice in the complex plane. Approximate representations of signals, using a truncated von Neumann lattice of only a few Gaussians, are considered. The error is quantified using various p-norms as accuracy measures, which reflect different practical needs. Optimization techniques are used to find optimal coefficients and to further reduce the size of the basis, whilst still preserving a good degree of accuracy. Examples are presented.

ANISOTROPY IN THE COMPRESSIBLE QUANTUM HALL STATE

Using a mean field theory on the von Neumann lattice, we study compressible anisotropic states around ν=l+1/2 in the quantum Hall system. The Hartree-Fock energy of the unidirectional charge density wave (UCDW) are calculated self-consistently. In these states the UCDW seems to be the most plausible state. We show that the UCDW is regarded as a collection of the one-dimensional lattice fermion systems which extend to the uniform direction. The kinetic energy of this one-dimensional system is induced from the Coulomb interaction term and the self-consistent Fermi surface is obtained.

Paired and stripe states in the quantum Hall system

We study a paired state at the half-filled Landau level using a mean field theory on the von Neumann lattice. We obtain a microscopic model which shows a continuous transition from the compressible stripe state to the paired state. The energy gap in the paired state is calculated numerically at the half-filled second Landau level.

Pressure, resistance, and current activation of anisotropic compressible Hall states

Thermodynamic and electric properties of anisotropic compressible Hall states at higher Landau levels are studied using a mean field theory on the von Neumann lattice basis. It is shown that resistances agree with the recent experiments of anisotropic compressible states and the states have negative pressure. As implications, the collapse phenomena of the integer quantum Hall effect are discussed.

Derivative Uniform Sampling via Weierstrass σ(z). Truncation Error Analysis in

Abstract In the entire functions space consisting of at most second order functions such that their type is less than πq/(2s 2) it is valid the q-order derivative sampling series reconstruction procedure, reading at the von Neumann lattice {s(m + ni)| (m, n) ∈ } via the Weierstrass σ(·) as the sampling function, s > 0. The uniform convergence of the sampling sums to the initial function is proved by the circular truncation error upper bound, especially derived for this reconstruction procedure. Finally, the explicit second and third order sampling formulæ are given.

Effective Hamiltonian for striped and paired states at the half-filled Landau level

We study a pairing mechanism for the quantum Hall system using a mean field theory with a basis on the von Neumann lattice, on which the magnetic translations commute. In the Hartree-Fock-Bogoliubov approximation, we solve the gap equation for spin-polarized electrons at the half-filled Landau levels. We obtain an effective Hamiltonian which shows a continuous transition from the compressible striped state to the paired state. Furthermore, a crossover occurs in the pairing phase. The energy spectrum and energy gap of the quasiparticle in the paired state is calculated numerically at the half-filled second Landau level.

Orthonormal coherent states on von Neumann lattices

An averaging procedure in phase plane is developed leading to orthonormality of coherent states on a von Neumann lattice. These states correspond to entire unit cells of area h (Planck constant) in the phase plane and they can be specified by (mb,n 2π/b), where m and n are integers and b is a constant related to the spread of the harmonic oscillator ground state. The product of uncertainties for the co-ordinate x and momentum p in these states is very close to . This is by a factor of 2π smaller than the unit cell area, and makes it, in principle, possible to measure simultaneously x and p for the location of a unit cell in the phase plane.

String scale in noncommutative Yang–Mills

We identify the effective string scale of noncommutative Yang-Mills theory (NCYM) with the noncommutativity scale through its dual supergravity description. We argue that Newton's force law may be obtained with 4 dimensional NCYM with maximal SUSY. It provides a nonperturbative compactification mechanism of IIB matrix model. We can associate NCYM with the von Neumann lattice by the bi-local representation. We argue that it is superstring theory on the von Neumann lattice. We show that our identification of its effective string scale is consistent with exact T-duality (Morita equivalence) of NCYM.

Gabor representation of optical signals using a truncated von Neumann lattice and its practical implementation

The Gabor expansion of an arbitrary optical signal in terms of Gaussian signals that form an N3M truncated von Neumann lattice in the time-frequency plane is studied. The accuracy of the expansion for various values of (N,M) and various widths of the Gaussians is exam- ined. The accuracy of the method is also demonstrated by comparing and contrasting the Wigner function for both the original and recon- structed signals. Practical implementation of the method through ''Gabor analyzers'' that directly measure the Gabor coefficients of optical signals is also discussed. © 2000 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(00)02507-1)

Compressible anisotropic states around the half-filled Landau levels

Using the von Neumann lattice formalism, we study compressible anisotropic states around the half-filled Landau levels in the quantum Hall system. In these states the unidirectional charge-density wave (UCDW) state seems to be the most plausible state. The charge-density profile and Hartree-Fock energy of the UCDW are calculated self-consistently. The wavelength dependence of the energy for the UCDW is also obtained numerically. We show that the UCDW is regarded as a collection of the one-dimensional lattice Fermi-gas systems which extend to the uniform direction. The kinetic energy of the gas system is generated dynamically from the Coulomb interaction.

Stability of the Compressible Quantum Hall State around the Half-Filled Landau Level

We study the compressible states in the quantum Hall system using a mean field theory on the von Neumann lattice. In the lowest Landau level, a kinetic energy is generated dynamically from Coulomb interaction. The compressibility of the state is calculated as a function of the filling factor $\nu$ and the width $d$ of the spacer between the charge carrier layer and dopants. The compressibility becomes negative below a critical value of $d$ and the state becomes unstable at $\nu=1/2$. Within a finite range around $\nu=1/2$, the stable compressible state exists above the critical value of $d$.

Field theory on the von Neumann lattice and the quantized Hall conductance of Bloch electrons

We construct useful sets of one-particle states in the quantum Hall system based on the von Neumann lattice. Using the set of momentum states, we develop a field-theoretical formalism and apply the formalism to the system subjected to a periodic potential. The topological formula of the Hall conductance written by the winding number of propagator is generalized to Bloch electrons. The relation between the winding number and the Chern number is clarified.

Construction of quantum states from an optimally truncated von Neumann lattice of coherent states

Arbitrary quantum states can be expanded with a very good accuracy in terms of coherent states on a truncated von Neumann lattice. Optimization techniques are used to further reduce the size of the basis, whilst still preserving the same degree of accuracy. Various p-norms are adopted as accuracy measures, which reflect different practical needs. Numerical examples demonstrate that a significant reduction in the size of the basis can be achieved with optimization techniques, without substantial loss of accuracy.

Theory of current-induced breakdown of the quantum Hall effect

By studying the quantum Hall effect of stationary states with high values of injected current using a von Neumann lattice representation, we found that broadening of extended state bands due to a Hall electric field occurs and causes the breakdown of the quantum Hall effect. The Hall conductance agrees with a topological invariant that is quantized exactly below a critical field and is not quantized above a critical field. The critical field is proportional to $B^{3/2}$ and is enhanced substantially if the extended states occupy a small fraction of the system.

Duality relation among periodic-potential problems in the lowest Landau level

Using a momentum representation of a magnetic von Neumann lattice, we study a two-dimensional electron in a uniform magnetic field and obtain one-particle spectra of various periodic short-range potential problems in the lowest Landau level.We find that the energy spectra satisfy a duality relation between a period of the potential and a magnetic length. The energy spectra consist of the Hofstadter-type bands and flat bands. We also study the connection between a periodic short-range potential problem and a tight-binding model.

Robust quantum state engineering using coherent states on a truncated von Neumann lattice

Quantum states are constructed as a superposition of a few coherent states on a truncated von Neumann lattice. It is shown that the reconstruction is robust, in the sense that random noise in the coefficients affects the constructed state weakly. Numerical results for squeezed states and number eigenstates demonstrate that the technique is very accurate.

Robust quantum state engineering using coherent states on a truncated von Neumann lattice

Quantum states are constructed as a superposition of a few coherent states on a truncated von Neumann lattice. It is shown that the reconstruction is robust, in the sense that random noise in the coefficients affects the constructed state weakly. Numerical results for squeezed states and number eigenstates demonstrate that the technique is very accurate.

Balian-Low Theorem for Landau Levels

The Balian-Low theorem is applied to the motion of an electron in the xy plane with a magnetic field B perpendicular to this plane. The energy spectrum for this problem is the Landau levels. It is shown that the eigenfunctions for the Landau levels cannot be chosen sufficiently localized in order to make both uncertainties $\ensuremath{\Delta}x$ and $\ensuremath{\Delta}y$ finite. A similar result holds for the coordinates of the orbit center. With this restriction on the localization, complete orthonormal sets are defined on von Neumann lattices.

The growth of Bargmann functions and the completeness of sequences of coherent states

The growth of Bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states. Using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of Bargmann functions of an order not exceeding (and of a type not exceeding if of order ). These results generalize known results on the completeness of von Neumann lattices. The practical significance of this formalism in the context of quantum optics is also discussed.

Flux State in von Neumann Lattices and the Fractional Hall Effect

Formulation of quantum Hall dynamics using von Neumann lattice of guiding center coordinates is presented. A topological invariant expression of the Hall conductance is given and a new mean field theory of the fractional Hall effect based on flux condensation is proposed. Because our mean field Hamiltonian has the same form as Hofstadter Hamiltonian, it is possible to understand characteristic features of the fractional Hall effect from Hofstadter's spectrum. Energy gap and other physical quantities are computed and are compared with the experiments. A reasonable agreement is obtained.