Effective Single-particle Research Articles

Fermionic and bosonic ac conductivities at strong disorder

We study the ac conduction in a system of fermions or bosons strongly localised in a disordered array of sites with short-range interactions at frequencies larger than the intersite tunnelling but smaller than the characteristic fluctuation of the on-site energy. While the main contribution $\sigma_0(\omega)$ to the conductivity comes from local dipole-type excitations on close pairs of sites, coherent processes on three or more sites lead to an interference correction $\sigma_1(\omega)$, which depends on the statistics of the charge carriers and can be suppressed by magnetic field. For bosons the correction is always positive, while for fermions it can be positive or negative depending on whether the conduction is dominated by effective single-particle or single-hole processes. We calculate the conductivity explicitly assuming a constant density of states of single-site excitations. Independently of the statistics, $\sigma_0(\omega)=const$. For bosons $\sigma_1(\omega)\propto \log(C/\omega)$. For fermions $\sigma_1(\omega)\propto\log[\max(A,\omega)/\omega]-\log[\max(B,\omega)/\omega]$, where the first and the second term are respectively the particle and hole contributions, $A$ and $B$ being the particle and hole energy cutoffs. The ac magnetoresistance has the same sign as $\sigma_1(\omega)$.

Calculations for charge transfer and ionization in heavy-particle collisions from water molecules

A recently introduced method for the description of bare ion-molecule collisions is extended to deal with intermediate-energy He+-H2O collisions, in which both the projectile electron and the target electrons are allowed to undergo transitions. All initially populated orbitals are propagated simultaneously using the same effective single-particle Hamiltonian to ensure their orthogonality at all times. The asymptotic solutions are combined to form many-electron transition probabilities on the level of single determinantal wave functions, thereby respecting the Pauli principle. It is shown that this analysis has interesting consequences for capture, loss, and ionization channels.

Semi-classical approximation for second-harmonic generation in nanoparticles

Second-harmonic generation by spherical nanoparticles is a non-local optical process that can also be viewed as the result of the nonlinear response of the a interface layer. The classical electrodynamic description, based e.g. on the nonlinear Mie theory, entails the knowledge of the dielectric function and the surface nonlinear optical susceptibility; both quantities are usually assumed to be predetermined, for instance from experiment. We propose here an approach based on the semi-classical approximation for the quantum sum-over-states expression that allows one to capture the second-order optical process from first principles. A key input is the electronic density, which can be obtained from effective single particle approaches such as density-functional theory in the local density implementation. We show that the resulting integral equations can be solved very efficiently rendering thus the treatment of macroscopic systems. As an illustration we present numerical results for the magic Na−2869 cluster.

Mesoscopic Anderson box: Connecting weak to strong coupling

Both the weakly coupled and strong coupling Anderson impurity problems are characterized by a Fermi-liquid theory with weakly interacting quasiparticles. In an Anderson box, mesoscopic fluctuations of the effective single particle properties will be large. We study how the statistical fluctuations at low temperature in these two problems are connected, using random matrix theory and the slave boson mean field approximation (SBMFA). First, for a resonant level model such as results from the SBMFA, we find the joint distribution of energy levels with and without the resonant level present. Second, if only energy levels within the Kondo resonance are considered, the distributions of perturbed levels collapse to universal forms for both orthogonal and unitary ensembles for all values of the coupling. These universal curves are described well by a simple Wigner-surmise type toy model. Third, we study the fluctuations of the mean field parameters in the SBMFA, finding that they are small. Finally, the change in the intensity of an eigenfunction at an arbitrary point is studied, such as is relevant in conductance measurements: we find that the introduction of the strongly-coupled impurity considerably changes the wave function but that a substantial correlation remains.

Dirac particle in a spherical scalar potential well

In this paper we investigate a solution of the Dirac equation for a spin-$\frac{1}{2}$ particle in a scalar potential well with full spherical symmetry. The energy eigenvalues for the quark particle in ${s}_{1/2}$ states (with $\ensuremath{\kappa}=\ensuremath{-}1$) and ${p}_{1/2}$ states (with $\ensuremath{\kappa}=1$) are calculated. We also study the continuous Dirac wave function for a quark in such a potential, which is not necessarily infinite. Our results, at infinite limit, are in good agreement with the MIT bag model. We make some remarks about the sharpness value of the wave function on the wall. This model, for finite values of potential, also could serve as an effective model for the nucleus where $U(r)$ is the effective single particle potential.

Quantum phase-space analysis of population equilibration in multiwell ultracold atomic systems

We examine the medium time quantum dynamics and population equilibration of two-, three-, and four-well Bose-Hubbard models using stochastic integration in the truncated Wigner phase-space representation. We find that all three systems will enter at least a temporary state of equilibrium, with the details depending on both the classical initial conditions and the initial quantum statistics. We find that classical integrability is not necessarily a good guide as to whether equilibration will occur. We construct an effective single-particle reduced density matrix for each of the systems, using the expectation values of operator moments, and use this to calculate an effective entropy. Knowing the expected maximum values of this entropy for each system, we are able to quantify the different approaches to equilibrium.

Creating exotic condensates via quantum-phase-revival dynamics in engineered lattice potentials

In the field of ultracold atoms in optical lattices a plethora of phenomena governed by the hopping energy $J$ and the interaction energy $U$ have been studied in recent years. However, the trapping potential typically present in these systems sets another energy scale and the effects of the corresponding time scale on the quantum dynamics have rarely been considered. Here we study the quantum collapse and revival of a lattice Bose-Einstein condensate (BEC) in an arbitrary spatial potential, focusing on the special case of harmonic confinement. Analyzing the time evolution of the single-particle density matrix, we show that the physics arising at the (temporally) recurrent quantum phase revivals is essentially captured by an effective single particle theory. This opens the possibility to prepare exotic non-equilibrium condensate states with a large degree of freedom by engineering the underlying spatial lattice potential.

Efficient Solutions of Self-Consistent Mean Field Equations for Dewetting and Electrostatics in Nonuniform Liquids

We use a new configuration-based version of linear response theory to efficiently solve self-consistent mean field equations relating an effective single particle potential to the induced density. The versatility and accuracy of the method is illustrated by applications to dewetting of a hard sphere solute in a Lennard-Jones fluid, the interplay between local hydrogen bond structure and electrostatics for water confined between two hydrophobic walls, and to ion pairing in ionic solutions. Simulation time has been reduced by more than an order of magnitude over previous methods.

Shell evolution and nuclear forces

We present a quantitative study of the role played by different components characterizing the nucleon–nucleon interaction in the evolution of the nuclear shell structure. It is based on the spin–tensor decomposition of an effective two-body shell-model interaction and the subsequent study of effective single-particle energy variations in a series of isotopes or isotones. The technique allows to separate unambiguously contributions of the central, vector and tensor components of the realistic effective interaction. We show that while the global variation of the single-particle energies is due to the central component of the effective interaction, the characteristic behavior of spin–orbit partners, noticed recently, is mainly due to its tensor part. Based on the analysis of a well-fitted realistic interaction in the sdpf shell-model space, we analyze in detail the role played by the different terms in the formation and/or disappearance of N=16, N=20 and N=28 shell gaps in neutron-rich nuclei.

A thermal self-consistent field theory for the calculation of molecular vibrational partition functions

A new approach for the calculation of anharmonic molecular vibrational partition functions is developed based on a separable ansatz to the thermal density matrix. The parameters appearing in the effective single particle Hamiltonians that generate the thermal density matrices are determined variationally. The resulting equations are the thermal analogs of the vibrational self-consistent field approximation. The method has the formal property that the free energy calculated by this approach is an upper bound to the exact free energy. Thermodynamic quantities calculated by this approach are generally in good agreement with the results of numerically converged calculations. This approach is more efficient than the standard sum over state approaches in that the computational resources scale with N(4) where N is the number of vibrational degrees of freedom. Thus it can be applied to fairly large systems.

Wigner energy and shell gaps in two-nucleon separation energies

Two-nucleon separation energies are differences of binding energies. They provide important information on the relative stability of nuclei and, in particular, on shell gaps. In this work, the behavior of the decrements (change of slope) of two-nucleon separation energies has been studied. It has been shown that the bulk of this decrement at a shell-gap consists mainly of twice the difference of two effective single particle energies, plus a pairing correction. The decrement of the two-nucleon separation energies has a maximum (spike) for $N=Z$ nuclei. A comparison with values calculated by using a seniority binding energy formula shows that the spike is due to the Wigner energy. The evolution with nuclear mass of the isospin dependence of the Wigner energy is discussed.

Insulator-metal transition in biased finite polyyne systems

A method for the study of the electronic transport in strongly coupled electron-phonon systems is formalized and applied to a model of polyyne chains biased through metallic Au leads. We derive a stationary non equilibrium polaronic theory in the general framework of a variational formulation. The numerical procedure we propose can be readily applied if the electron-phonon interaction in the device hamiltonian can be approximated as an effective single particle electron hamiltonian. Using this approach, we predict that finite polyyne chains should manifest an insulator-metal transition driven by the non-equilibrium charging which inhibits the Peierls instability characterizing the equilibrium state.

Linear response for confined particles

The dynamics of fluctuations is considered for electrons near a positive ion or for charges in a confining trap. The stationary nonuniform equilibrium densities are discussed and contrasted. The linear response function for small perturbations of this nonuniform state is calculated from a linear Markov kinetic theory whose generator for the dynamics is exact in the short time limit. The kinetic equation is solved in terms of an effective mean field single particle dynamics determined by the local density and dynamical screening by a dielectric function for the nonuniform system. The autocorrelation function for the total force on the charges is discussed.

Reverse engineering in many-body quantum physics: Correspondence between many-body systems and effective single-particle equations

The mapping, exact or approximate, of a many-body problem onto an effective single-body problem is one of the most widely used conceptual and computational tools of physics. Here, we propose and investigate the inverse map of effective approximate single-particle equations onto the corresponding many-particle system. This approach allows us to understand which interacting system a given single-particle approximation is actually describing, and how far this is from the original physical many-body system. We illustrate the resulting reverse engineering process by means of the Kohn-Sham equations of density-functional theory. In this application, our procedure sheds light on the nonlocality of the density-potential mapping of density-functional theory, and on the self-interaction error inherent in approximate density functionals.

Self-consistent study of Anderson localization in the Anderson-Hubbard model in two and three dimensions

We consider the change in electron localization due to the presence of electron-electron repulsion in the \HA model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. We find rich nonmonotonic behavior of the localization length $\xi$ in two-dimensional systems, including an interaction-induced exponential enhancement of $\xi$ for small and intermediate disorders although $\xi$ remains finite. In three dimensions we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbard-gapped phases. For small $U$ there is re-entrant behavior from the Anderson localized phase to the metallic phase.

Generalized Kohn-Sham system in one-matrix functional theory

A system of electrons in a local or nonlocal external potential can be described with one-matrix functional theory (1MFT), which is similar to density-functional theory (DFT) but takes the one-particle reduced density matrix (one-matrix) instead of the density as its basic variable. Within 1MFT, Gilbert derived [Phys. Rev. B 12, 2111 (1975)] effective single-particle equations analogous to the Kohn-Sham (KS) equations in DFT. The self-consistent solution of these 1MFT-KS equations reproduces not only the density of the original electron system but also its one-matrix. While in DFT it is usually possible to reproduce the density using KS orbitals with integer (0 or 1) occupancy, in 1MFT reproducing the one-matrix requires in general fractional occupancies. The variational principle implies that the KS eigenvalues of all fractionally occupied orbitals must collapse at self-consistency to a single level. We show that as a consequence of the degeneracy, the iteration of the KS equations is intrinsically divergent. Fortunately, the level-shifting method, commonly introduced in Hartree-Fock calculations, is always able to force convergence. We introduce an alternative derivation of the 1MFT-KS equations that allows control of the eigenvalue collapse by constraining the occupancies. As an explicit example, we apply the 1MFT-KS scheme to calculate the ground state one-matrix of an exactly solvable two-site Hubbard model.

Direct energy functional minimization under orthogonality constraints

The direct energy functional minimization problem in electronic structure theory, where the single-particle orbitals are optimized under the constraint of orthogonality, is explored. We present an orbital transformation based on an efficient expansion of the inverse factorization of the overlap matrix that keeps orbitals orthonormal. The orbital transformation maps the orthogonality constrained energy functional to an approximate unconstrained functional, which is correct to some order in a neighborhood of an orthogonal but approximate solution. A conjugate gradient scheme can then be used to find the ground state orbitals from the minimization of a sequence of transformed unconstrained electronic energy functionals. The technique provides an efficient, robust, and numerically stable approach to direct total energy minimization in first principles electronic structure theory based on tight-binding, Hartree-Fock, or density functional theory. For sparse problems, where both the orbitals and the effective single-particle Hamiltonians have sparse matrix representations, the effort scales linearly with the number of basis functions N in each iteration. For problems where only the overlap and Hamiltonian matrices are sparse the computational cost scales as O(M2N), where M is the number of occupied orbitals. We report a single point density functional energy calculation of a DNA decamer hydrated with 4003 water molecules under periodic boundary conditions. The DNA fragment containing a cis-syn thymine dimer is composed of 634 atoms and the whole system contains a total of 12,661 atoms and 103,333 spherical Gaussian basis functions.

Remarkable microstructure evolution in biogenic crystals upon mild annealing

24 European Crystallographic Meeting, ECM24, Marrakech, 2007 Page s77 Acta Cryst. (2007). A63, s77 average structure reveals the distribution of the ions as described by the probability density function, which can be obtained either by Fourier transformation of the Debye Waller factor (including anharmonic terms [1]) or by Fourier methods. The former is more appropriate for interstitial diffusion processes, i.e. when the migration of the ions proceeds directly to adjacent vacant sites. In this case a continuous density is found directly representing the diffusion pathway. Moreover, by applying Boltzmann statistics effective single particle potentials can be derived containing the potential barriers to migration which can be compared with otherwise measured activation energies. On the other hand, if interstitialcy (exchange) processes prevail, it is more meaningful to analyse difference Fourier maps and/or to introduce additional (metastable) positions. All this will be illustrated by various examples. For fluorite like ZrO2 doped with cations and anions (e.g. Sc and N [2]) one generally finds anion diffusion pathways directly through an edge of the surrounding cation tetrahedron along to a neighbouring site. Derived activation energies agree with those obtained from conductivity measurements showing that no additional energy is needed for the creation of defects. By comparing samples with and without N it is possible to derive separate activation energies for O and N. In perovskite like LaGaO3 doped with Mg and Sr the pathways are curved. These and the corresponding potentials are significantly altered when measured under electric fields [3] or microwave irradiation [4]. An example of interstitialcy diffusion is provided by mayenite (Ca12Al14O33) [5]. Its structure may be described as a calciumaluminate framework containing 32 of the 33 O atoms, while the remaining “free” oxygen statistically occupies 1/6 of larger cages in the structure. In spite of the presence of vacancies and rather large openings between adjacent cages no continuous density was found, therewith ruling out interstitial diffusion (probably because of too long jump distances). On the other hand, difference Fourier maps revealed various other weakly occupied positions which can be related to an exchange process with particular framework O. Moreover, each jump process is connected with a relaxation of Ca, i.e. there is considerable interaction with the framework. Interestingly, the mechanism is different when the “free” O is replaced by N, i.e. nitrogen shows the interstitial type of diffusion within the same framework.

Disentangling the effective single-particle gap in electron-doped cuprates: Electronic Raman absorption

Disentangling the effective single-particle gap in electron-doped cuprates: Electronic Raman absorption

Attraction Between Like-Charged Walls: Short-Ranged Simulations Using Local Molecular Field Theory

Effective attraction between like-charged walls mediated by counterions is studied using local molecular field (LMF) theory. Monte Carlo simulations of the "mimic system" given by LMF theory, with short-ranged "Coulomb core" interactions in an effective single particle potential incorporating a mean-field average of the long-ranged Coulomb interactions, provide a direct test of the theory, and are in excellent agreement with more complex simulations of the full Coulomb system by Moreira and Netz [Eur. Phys. J. E 8, 33 (2002)]. A simple, generally applicable criterion to determine the consistency parameter sigma(min) needed for accurate use of the LMF theory is presented.