Kohn Theorem Research Articles

Spurious modes in Extended RPA theories

Necessary conditions that the spurious state associated with the translational motion and its double-phonon state have zero excitation energy in extended RPA (ERPA) theories which include both one-body and two-body amplitudes are investigated using the small amplitude limit of the time-dependent density-matrix theory (STDDM). STDDM provides us with a quite general form of ERPA as compared with other similar theories in the sense that all components of one-body and two-body amplitudes are taken into account. Two conditions are found necessary to guarantee the above property of the single and double spurious states: The first is that no truncation in the single-particle space should be made. This condition is necessary for the closure relation to be used and is common for the single and double spurious states. The second depends on the mode. For the single spurious state all components of the one-body amplitudes must be included, and for the double spurious state all components of one-body and two-body amplitudes have to be included. It is also shown that the Kohn theorem and the continuity equations for transition densities and currents hold under the same conditions as the spurious states. ERPA theories formulated using the Hartree-Fock ground state have a non-hermiticity problem. A method for formulating ERPA with hermiticity is also proposed using the time-dependent density-matrix formalism.

Corollary to the Hohenberg–Kohn theorem

AbstractAccording to the Hohenberg–Kohn theorem, there is an invertible one‐to‐one relationship between the Hamiltonian of a system and the corresponding ground‐state density ρ(r). The extension of the theorem to the time‐dependent case by Runge and Gross states that there is an invertible one‐to‐one relationship between the density ρ(rt) and the Hamiltonian (t). In the proof of the theorem, Hamiltonians /(t) that differ by an additive constant C/function C(t) are considered equivalent. Because the constant C/function C(t) is extrinsically additive, the physical system defined by these differing Hamiltonians /(t) is the same. Thus, according to the theorem, the density ρ(r)/ρ(rt) uniquely determines the physical system as defined by its Hamiltonian /(t). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians {}/{(t)} that differ by an intrinsic constant C/function C(t) but which represent different physical systems and yet possess the same density ρ(r)/ρ(rt). The intrinsic constant C/function C(t) contains information about the different physical systems and helps differentiate between them. In such a case, the density ρ(r)/ρ(rt) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians {}/{(t)}. Thus, although the proof of the Hohenberg–Kohn theorem is independent of whether the constant C/function C(t) is additive or intrinsic, the applicability of the theorem is restricted to excluding the case of the latter. The corollary is as follows: degenerate Hamiltonians {}/{(t)} that represent different physical systems, but differ by a constant C/function C(t), and yet possess the same density ρ(r)/ρ(rt), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Analysis of a General Theorem Concerning Two Non-Commuting Hermitian Matrices: Quantum Mechanical Implications for Ground and Excited States

A general theorem concerning the spectral relationship of two non-commuting Hermitian matrices is demonstrated. Discussion and analysis of such finding leads to consider its tight connection with respect of the Hohenberg–Kohn theorem (HKT), cornerstone of DFT theory. The present analysis shows that not only HKT can be considered a particular case of the proposed theorem, but also the validity of the studied spectral relationship can be extended from quantum mechanical ground state to excited states as well.

Low-energy excitations of double quantum dots in the lowest Landau level regime

We study the spectrum and magnetic properties of double quantum dots in the lowest Landau level for different values of the hopping and Zeeman parameters by means of exact diagonalization techniques in systems of N=6 and N=7 electrons and filling factor close to 2. We compare our results with those obtained in double quantum layers and single quantum dots. The Kohn theorem is also discussed.

Excitations Below the Kohn Mode; FIR-Absorption in Quantum Dots

We have investigated the far-infrared (FIR) responses of an array of quantum dots in modulation doped GaAs heterostructures in an external magnetic field. In accordance with the extended Kohn theorem FIR-radiation can only excite the center-of-mass modes of harmonically confined electrons in a dot. In a perpendicular magnetic field the single absorption peak corresponding to the Kohn mode splits in two peaks separated by the cyclotron frequency. We observe novel modes at frequencies below the higher frequency Kohn mode. Comparison with Hartree-RPA calculations show that these modes arise from the flattened confining potential in our field-effect quantum dots. They reflect pronounced relative motion of the charge density with respect to its center-of-mass.

Zero-sound density oscillations in Fermi-Bose mixtures

Within a mean field plus random-phase approximation formalism, we investigate the collective excitations of a three-component Fermi-Bose mixture of K atoms, magnetically trapped and subjected to repulsive s-wave interactions. We analyze both the single-particle excitation and the density oscillation spectra created by external multipolar fields for varying fermion concentrations. The formalism and the numerical output are consistent with the generalized Kohn theorem for the whole multispecies system. The calculations give rise to fragmented density excitation spectra of the fermion sample and illustrate the role of the mutual interaction in the observed deviations of the bosonic spectra with respect to Stringari's rule.

Superconducting vortices in ac fields: does the Kohn theorem work?

Electrodynamics of clean pinning-free type II superconductors in the mixed state is derived using the Boltzmann kinetic equations for excitations. The condition of the vortex cyclotron resonance is found. The reason why this resonance does not comply with the Kohn theorem is discussed.

Density functional theory of magnetic systems revisited

The Hohenberg–Kohn theorem of density functional theory (DFT) for the case of electrons interacting with an external magnetic field (that couples to spin only) is examined in more detail than previously. An unexpected generalization is obtained: in certain cases (which include half-metallic ferromagnets and magnetic insulators), the ground state, and hence the spin density matrix, is invariant for some non-zero range of a shift in uniform magnetic field. The energy gap in an insulator or a half-metal is shown to be a ground state property of the N-electron system in magnetic DFT. Its relation to the gap in the Kohn–Sham eigenvalue spectrum is analyzed.

Kohn mode for trapped Bose gases within the dielectric formalism

The presence of undamped harmonic center of mass oscillations of a weakly interacting Bose gas in a harmonic trap is demonstrated within the dielectric formalism for a previously introduced finite temperature approximation including exchange. The consistency of the approximation with the Kohn theorem is thereby demonstrated. The Kohn modes are found explicitly, generalizing an earlier zero-temperature result found in the literature. It is shown how the Kohn mode disappears from the single-particle spectrum, while remaining in the density oscillation spectrum, when the temperature increases from below to above the condensation temperature.

Excitation spectra of harmonic quantum dot lattices with Coulomb interaction between the dots and the broken generalized Kohn theorem

Lattices of parabolic quantum dots with different dot species per unit cell and Coulomb interaction between the dots are investigated. As examples, we solve the Schr\odinger equation for square lattices with two different dots per unit cell: (i) two different circular dots, and (ii) two elliptical dots, which are rotated by $90\ifmmode^\circ\else\textdegree\fi{}$ relative to each other. The interaction between the dots is considered in a dipole approximation, and excitation spectra are calculated. For vanishing momentum transfer $(\mathbf{q}=0),$ the energy spectrum of the first case can be expressed as a superposition of two noninteracting dots with an effective confinement frequency, which includes the effect of dot interaction. Only in the second case is there a splitting of degenerate absorption lines, and an anticrossing occurs, which is a qualitative indication of interdot interaction. If the interaction becomes very strong and if all lattice sites (not necessarily confinement potentials) are equivalent, then the contribution of the dot interaction outweighs possible differences in the confinement potentials and the generalized Kohn theorem gradually reenters, in the sense that one pair of excitation modes (pseudo-Kohn modes) becomes independent of the interaction strength. For finite momentum transfer $(\mathbf{q}\ensuremath{\ne}0),$ we investigated mode softening and the influence of changing the interaction strength between dots of different sublattices. The latter effect may be implemented by putting different electron numbers in different dot species. It is shown that strengthening the next-nearest-neighbor interaction versus the nearest-neighbor interaction stabilizes the square lattice.

Charge redistribution effect on the properties of charge transfer complexes HnR?XY and HnR?X2 (X, Y = F, Cl, Br, I; R = O, S, N, P)

Systematic studies on structures, energies, charge transfer, dipole moments, and ionic character of a series of weakly bonded charge transfer (CT) complexes (D⋅AB, D = H2O, H2S, NH3, PH3, AB = F2, Cl2, Br2, I2, BrCl, IBr, ClF, ICl, BrF, IF) have been carried out by the hybrid Hartree–Fock density functional theory (HF-DFT) method, where those results are validated by available experimental and theoretical investigations. Employing the Hohenberg–Kohn theorem, the property of a multicomponent system is formulated with contributions from both component properties and the charge redistribution (CR) effect, which describes the electronic coupling between components. For any property of a multicomponent system, provided that the intercomponent coupling is weak enough, the first-order approximation can be applied, which yields a linear correlation of the component contribution to the CR effect. In fact, this kind of linear relationship can be evidenced by all the studied properties including the geometry, energy, charge transfer, dipole moment, and ionic character of all 40 complexes. This approximation quantitatively describes the relative contribution of the components to a given property, which shows the same tendency in a series of complexes. Based on the investigations of the CT effect on the intermolecular bond energy and the total dipole moment, it has been found that the principal bonding character of the title complexes was ascertained to be ionic with the exception of the F2 complexes, which agrees well with the calculated ionic character. The CT effect, though small in a quantitative aspect, is directly connected to various kinds of system properties. The effectiveness and consistency of the present type of calculations in multicomponent systems may allow their wider applications in the study of intermolecular interactions. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 327–338, 2001

Symmetry and degeneracy in density functional theory

AbstractThe superhamiltonian formalism is used to construct a totally symmetric superdensity for a system whose ground state corresponds to a multidimensional irreducible representation of the symmetry group of the Hamiltonian. A Hohenberg–Kohn theorem is proven and the Levy constrained‐search procedure is adapted. An appropriate generalization allows the application of this procedure to degenerate excited states. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

Solution of the Schrödinger equation for quantum-dot lattices with Coulomb interaction between the dots

The Schr\"odinger equation for quantum dot lattices with non-cubic, non-Bravais lattices built up from elliptical dots is investigated. The Coulomb interaction between the dots is considered in dipole approximation. Then only the center of mass (c.m.) coordinates of different dots couple with each other. This c.m. subsystem can be solved exactly and provides magneto- phonon like collective excitations. The inter-dot interaction is involved only through a single interaction parameter. The relative coordinates of individual dots form decoupled subsystems giving rise to intra-dot excitations. As an example, the latter are calculated exactly for two-electron dots. Emphasis is layed on qualitative effects like: i) Influence of the magnetic field on the lattice instability due to inter-dot interaction, ii) Closing of the gap between the lower and the upper c.m. mode at B=0 for elliptical dots due to dot interaction, and iii) Kinks in the single dot excitation energies (versus magnetic field) due to change of ground state angular momentum. It is shown that for obtaining striking qualitative effects one should go beyond simple cubic lattices with spherical dots. We also prove a more general version of the Kohn Theorem for quantum dot lattices. It is shown that for observing effects of electron- electron interaction between the dots in FIR spectra (breaking Kohn's Theorem) one has to consider dot lattices with at least two dot species with different confinement tensors.

Depolarization shift in coupled quantum wells with tunable level spectrum

The degeneracy of electronic subband levels in spike-inserted wide parabolic quantum wells is investigated by far-infrared spectroscopy and magnetotransport measurements. A narrow potential barrier within the parabolic well divides the electron system into two well-separated but strongly coupled layers, leading to a drastical change of the subband system and the collective intersubband excitations. In contrast to a single robust resonance, which, obeying Kohn's Theorem is usually seen in a purely parabolic quantum well, the collective intersubband transitions now recover the complex coupling and splitting of the single-particle states. By applying two different experimental methods being sensitive to either the single-particle spectrum or the collective excitation spectrum, we are able to confirm the theoretical predictions that the size of the observed splitting caused by resonant subband coupling is not altered by collective effects. Rather, the corresponding intersubband resonance lines are shifted to higher energies in a depolarization like manner.

Optically controlled confined potential of a vertically aligned double-dot array system

In a vertically coupled dot, it is expected that generalized Kohn theorem is invalid. Then electron–electron interaction can be observed in far-infrared absorption. This interaction depends on the coupling between dots. We fabricated a vertically aligned double-dot array using a double heterostructure to investigate the coupling. The strength of the coupling depends on the confine potential of the dot. In far-infrared magneto-optical absorption measurements under pulsed photoexcitation, we observed peak shift of asymmetry states and increasing of absorption intensity of symmetry states of the dot. This is a result of the confined potential modulation by light. This indicates the possibility of optical control of the coupling.

Magneto-optical Terahertz emission from plasmons in parabolic quantum wells

We report far-infrared emission from coupled plasmons in a parabolically graded AlGaAs/GaAs single quantum well driven by an inplane electric field. A magnetic field tilted at different angles with respect to the sample's growth axis is applied to couple the intersubband resonance to the cyclotron resonance. The measured mode dispersion shows an anticrossing as calculated in a simple centre of mass picture and confirms the Kohn theorem. For measured optical powers up to 10-7 W temperatures of the hot electron gas up to 70 K are found. At inplane magnetic fields a small satellite peak indicates deviations from ideal parabolicity.

Collective excitation frequencies of bosons in a parabolic potential with interparticle harmonic interactions

The fact that the ground-state first-order density matrix for bosons in a parabolic potential with interparticle harmonic interactions is known in exact form is here exploited to study collective excitations in the weak-coupling regime. Oscillations about the ground-state density are treated analytically by a linearized equation of motion which includes a kinetic energy contribution. We show that the dipole mode has the frequency of the bare trap, in accord with the Kohn theorem, and derive explicit expressions for the frequencies of the higher-multipole modes in terms of a frequency renormalized by the interactions.

Electromagnetic absorption of a pinned Wigner crystal at finite temperatures

We investigate the microwave absorption of a pinned, two-dimensional Wigner crystal in a strong magnetic field at finite temperatures. Using a model of a uniform commensurate pinning potential, we analyze thermal broadening of the electromagnetic absorption resonance. Surprisingly, we find that the pinning resonance peak should remain sharp even when the temperature is comparable or greater than the peak frequency. This result agrees qualitatively with recent experimental observations of the ac conductivity in two-dimensional hole systems in a magnetically induced insulating state. It is shown, in analogy with Kohn's theorem, that the electron-electron interaction does not affect the response of a harmonically pinned Wigner crystal to a spatially uniform external field at any temperature. We thus focus on anharmonicity in the pinning potential as a source of broadening. Using a $1/N$ expansion technique, we show that the broadening is introduced through the self-energy corrections to the magnetophonon Green's functions.

Electron spin resonance of the two-dimensional electron system inAlxGa1−xAs/GaAsat subunity filling factors

The electron spin resonance (ESR) of a two-dimensional electron system (2DES) in ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ in the regime of fractional filling (\ensuremath{\nu}1) of the lowest Landau level is investigated by a photoconductivity technique at millimeter wave frequencies. By performing the experiments in a standing wave the ESR is shown to be a magnetic-dipole transition. This is in agreement with a calculation based on an 8-band model and k\ensuremath{\cdot}p theory. The (single-electron) theory also yields excellent agreement for the experimental and theoretical magnetic field dependences of the ESR transition energy, i.e., single-electron transition energies are measured and Kohn's theorem is not violated. Nevertheless, due to electron-electron interaction the ground state of the 2DES can be a many-particle state with similar amplitudes of spin-up and spin-down contributions. Evidence for such a reduced spin polarization is found in a striking reduction of the ESR strength when decreasing the temperature from 1.6 to 0.3 K.

Far infrared response of InAs–GaSb type-II quantum dots

We have investigated theoretically the ground state and electromagnetic response of InAs–GaSb type-II quantum dots in the presence of a vertical magnetic field. The ground state is calculated within the Hartree approximation using a 2-band k· p effective mass Hamiltonian to represent the coupling between the two spin-split sets of dot levels derived from the lowest InAs electron subband and the highest GaSb heavy hole subband. The FIR response is calculated within the RPA. We show that the interlayer Coulomb interaction between the electrons and the holes combined with the k· p mixing breaks the generalized Kohn's theorem leading to a complex FIR spectrum.