Papers by Brandon van Zyl
Physical Review B, 2000
We have investigated the magnetoplasmon excitations in arrays of circular and noncircular quantum... more We have investigated the magnetoplasmon excitations in arrays of circular and noncircular quantum dots within the Thomas-Fermi-Dirac-von Weizsäcker approximation. Deviations from the ideal collective excitations of isolated parabolically confined electrons arise from local perturbations of the confining potential as well as interdot Coulomb interactions. The latter are unimportant unless the interdot separations are of the order of the size of the dots. Local perturbations such as radial anharmonicity and noncircular symmetry lead to clear signatures of the violation of the generalized Kohn theorem. In particular, the reduction of the local symmetry from SO(2) to C 4 results in a resonant coupling of different modes and an observable anticrossing behaviour in the power absorption spectrum. Our results are in good agreement with recent far-infrared (FIR) transmission experiments.
Physical Review B, 1994
A hydrodynamic description of the collective excitations of an inhomogeneous electronic system is... more A hydrodynamic description of the collective excitations of an inhomogeneous electronic system is developed on the basis of the Thomas-Fermi-Diracvon Weizsacker approximation to the equilibrium ground state. This approximation allows one to define realistic equilibrium densities which are then used to obtain a consistent description of the dynamical behavior. An application to a parabolically confined electron gas is presented and the magnetoplasmon modes are obtained from a solution of the linearized hydrodynamic equations. The wave-vector dispersion of the modes is determined, as well as the detailed dependence on the orientation of the applied magnetic field. The power absorption in the long-wavelength limit is also calculated to illustrate the center-of-mass mode excitations probed by transmission experiments. I. INTRODUC. T1ON A hydrodynamic description of electron dynamics in matter goes back to the early work of Bloch' and has since been applied to a wide range of problems. Within this approach, the many-electron system is represented as a charged fluid whose dynamics is described in terms of a density and velocity field. Its main appeal is its relative simplicity. Although the approach is usually introduced heuristically, with no presumption of theoretical rigor, it can be viewed as an approximate extension of density-functional theory to the dynamic regime. Considering the complexity of treating dynamics with formal many-body techniques, ' it is clearly of considerable interest to develop methods which are easier to implement and, at the same time, trustworthy in their qualitative predictions. The application to homogeneous systems is straightforward and in the case of an electron gas yields the expected collective plasmon excitation. This mode is sustained by electron-electron interactions which are accounted for in terms of a self-consistent polarization field generated by the density Quctuation. The internal properties of the support this kind of behavior. ' One is left with the im
Journal of Physics B Atomic Molecular and Optical Physics
We investigate the behavior of a dilute quasi two-dimensional, harmonically confined, weakly inte... more We investigate the behavior of a dilute quasi two-dimensional, harmonically confined, weakly interacting Bose gas within the finite-temperature Thomas-Fermi approximation. We find that the thermodynamic properties of the system are markedly different for repulsive and attractive interactions. Specifically, in contrast to the repulsive case, there appears to be a phase transition when the atoms interact with an attractive pseudo-potential, in the sense that there is no self-consistent solution for the normal ground state below a certain temperature T ⋆. These numerical findings are supported by analytical investigations of the thermodynamics of the system in the complex fugacity plane, and within the random-phase approximation. We also show that the temperature T ⋆ can be interpreted as the limiting temperature below which the system cannot be described as a collection of noninteracting haldons.
Physical Review E, 2003
Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zer... more Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one-dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D = 1.5 for the Riemann zeros and D = 1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbour spacings of the Riemann zeros are known to be chaotically distributed whereas the primes obey almost poisson-like statistics. Our findings show that the fractal dimension is dependent on both the level-statistics and spectral rigidity, ∆3, of the energy levels.
Physical Review A, 2004
We present a Hartree-Fock-Bogoliubov (HFB) theoretical treatment of the two-dimensional trapped B... more We present a Hartree-Fock-Bogoliubov (HFB) theoretical treatment of the two-dimensional trapped Bose gas and indicate how semiclassical approximations to this and other formalisms have lead to confusion. We numerically obtain results for the quantum mechanical HFB theory within the Popov approximation and show that the presence of the trap stabilizes the condensate against long wavelength fluctuations. These results are used to show where phase fluctuations lead to the formation of a quasicondensate.
Physical Review B, 2011
We examine the leading order semiclassical gradient corrections to the non-interacting kinetic en... more We examine the leading order semiclassical gradient corrections to the non-interacting kinetic energy density functional of a two dimensional Fermi gas by applying the extended Thomas-Fermi theory at finite temperature. We find a non-zero von Weizsäcker-like gradient correction, which in the high-temperature limit, goes over to the functional form (2 /24m)(∇ρ) 2 /ρ. Our work provides a theoretical justification for the inclusion of gradient corrections in applications of density-functional theory to inhomogeneous twodimensional Fermi systems at any finite temperature.
Physical Review Letters, 2001
We derive simple analytical expressions for the particle density ρ(r) and the kinetic energy dens... more We derive simple analytical expressions for the particle density ρ(r) and the kinetic energy density τ (r) for a system of noninteracting fermions in a d-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or localdensity) approximation for the functional relation τ [ρ] using the exact ρ(r) and show that it locally reproduces the exact kinetic energy density τ (r), including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of τT F [ρ(r)] yields the exact total kinetic energy.
Physical Review E, 2008
Prime numbers are the building blocks of our arithmetic, however, their distribution still poses ... more Prime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the non-trivial zeros of the Riemann ζ(s) function. According to the Hilbert-Pólya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of ζ(s). This idea encourages physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Marčhenko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the ζ(s) function. We demonstrate the multifractal nature of these potentials by measuring the Rényi dimension of their graphs. Our results offer hope for further analytical progress.
Physical Review B, 2004
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, twod... more We present closed-form, analytical expressions for the thermodynamic properties of an ideal, twodimensional ͑2D͒ charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas ͑in which case our expressions are exact͒ and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the currentdensity-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas ͑CBG͒ in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature T c à , below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for TϽT c à and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.
Physical Review B, 2006
An analytical expression for the first-order density matrix of a charged, two-dimensional, harmon... more An analytical expression for the first-order density matrix of a charged, two-dimensional, harmonically confined quantum gas, in the presence of a constant magnetic field is derived. In contrast to previous results available in the literature, our expressions are exact for any temperature and magnetic field strength. We also present a novel factorization of the Bloch density matrix in the form of a simple product with a clean separation of the zero-field and field-dependent parts. This factorization provides an alternative way of analytically investigating the effects of the magnetic field on the system, and also permits the extension of our analysis to other dimensions, and/or anisotropic confinement.
Physical Review A, 2003
We present closed analytical expressions for the particle and kinetic energy spatial densities at... more We present closed analytical expressions for the particle and kinetic energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic oscillator potential. For d = 2 and 3, exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the l−degeneracy in a harmonic oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.
Physical Review A, 2013
We systematically develop a density functional description for the equilibrium properties of a tw... more We systematically develop a density functional description for the equilibrium properties of a two-dimensional, harmonically trapped, spin-polarized dipolar Fermi gas based on the Thomas-Fermi von Weizsäcker approximation. We pay particular attention to the construction of the twodimensional kinetic energy functional, where corrections beyond the local density approximation must be motivated with care. We also present an intuitive derivation of the interaction energy functional associated with the dipolar interactions, and provide physical insight into why it can be represented as a local functional. Finally, a simple, and highly efficient self-consistent numerical procedure is developed to determine the equilibrium density of the system for a range of dipole interaction strengths.
Physical Review A, 2014
The average-density approximation is used to construct a nonlocal kinetic energy functional for a... more The average-density approximation is used to construct a nonlocal kinetic energy functional for an inhomogeneous two-dimensional Fermi gas. This functional is then used to formulate a Thomas-Fermi von Weizsäcker-like theory for the description of the ground state properties of the system. The quality of the kinetic energy functional is tested by performing a fully self-consistent calculation for an ideal, harmonically confined, two-dimensional system. Good agreement with exact results are found, with the number and kinetic energy densities exhibiting oscillatory structure associated with the nonlocality of the energy functional. Most importantly, this functional shows a marked improvement over the two-dimensional Thomas-Fermi von Weizsäcker theory, particularly in the vicinity of the classically forbidden region.
Laser Physics Letters, 2008
We investigate a density-functional theory (DFT) approach for an unpolarized trapped dilute Fermi... more We investigate a density-functional theory (DFT) approach for an unpolarized trapped dilute Fermi gas in the unitary limit. A reformulation of the recent work of T. Papenbrock [Phys. Rev. A, 72, 041602(R) (2005)] in the language of fractional exclusion statistics allows us to obtain an estimate of the universal factor, ξ 3D , in three dimensions (3D), in addition to providing a systematic treatment of finite-N corrections. We show that in 3D, finite-N corrections lead to unphysical values for ξ 3D , thereby suggesting that a simple DFT applied to a small number of particles may not be suitable in 3D. We then perform an analogous calculation for the two-dimensional (2D) system in the infinite-scattering length regime, and obtain a value of ξ 2D = 1. Owing to the unique properties of the Thomas-Fermi energy density-functional in 2D our result, in contrast to 3D, is exact and therefore requires no finite-N corrections.
Journal of Physics A: Mathematical and Theoretical, 2009
Motivated by the recent article of P. Shea et al. [Am. J. Phys. 77 (6), 2009] we examine the exac... more Motivated by the recent article of P. Shea et al. [Am. J. Phys. 77 (6), 2009] we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting via a short range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, provided that the range of the potential is much smaller than the oscillator length, in addition to clearly illustrating why regularization is not required in the limit of zero range. The two dimensional problem is less trivial, requiring a more careful treatment as compared to the one dimensional case. Our two dimensional analysis likewise reveals that the low-energy physics is also universal, in addition to providing a simple method for obtaining the appropriately regularized two dimensional pseudopotential.
Journal of Physics A: Mathematical and Theoretical, 2008
We present exact analytical results for the thermodyanmic properties of a two-dimensional (2D), h... more We present exact analytical results for the thermodyanmic properties of a two-dimensional (2D), harmonically trapped charged quantum gas in a magnetic field. While our results are applicable to both Fermi and Bose gases, we focus our attention on trapped fermions owing to their relevant application in the density-functional theory of inhomogeneous Fermi systems. In particular, we test the Thomas-Fermi (or continuum) approximation (TFA) for the functional relation tau[rho] using the exact rho(r) and show that it reproduces the local and global properties of the exact kinetic energy density tau(r) surprisingly well. However, when we compare our exact results for various thermodynamic quantities with the TFA, we find that it misses several important features. These deviations are shown to be entirely due to the quantum mechanical properties of the system, which are not accounted for in the continuum approximation.
Journal of Physics A: Mathematical and Theoretical, 2012
We investigate the Zel'dovich effect in the context of ultra-cold, harmonically trapped quantum g... more We investigate the Zel'dovich effect in the context of ultra-cold, harmonically trapped quantum gases. We suggest that currently available experimental techniques in cold-atoms research offer an exciting opportunity for a direct observation of the Zel'dovich effect without the difficulties imposed by conventional condensed matter and nuclear physics studies. We also demonstrate an interesting scaling symmetry in the level rearragements which has heretofore gone unnoticed.
Journal of Physics A: Mathematical and Theoretical, 2013
EPL (Europhysics Letters), 2010
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Papers by Brandon van Zyl