Two-particle Interaction Research Articles

Extended dynamical mean-field theory andGWmethod

We develop the extended dynamical mean field theory (E-DMFT) with a view towards realistic applications. {\bf 1)} We introduce an intuitive derivation of the E-DMFT formalism. By identifying the Hartree contributions before the E-DMFT treatment, it allows to handle systems in symmetry breaking phases within a simple formalism. {\bf 2)} We make a new implementation of E-DMFT through real Hubbard-Stratonovich transformation to decouple the non-local two-particle interactions. We apply it to a 3D U-V model and investigate the behavior of the various Green's functions, especially the density susceptibility, as the density instability is approached. We obtain the phase diagram at a finite temperature. {\bf 3)} We present a formalism incorporating E-DMFT with Cellular DMFT. {\bf 4)} We suggest an improvement of the E-DMFT approach by combining it with a generalized GW method. The method combines the local self-energy from E-DMFT and the non-local ones from the perturbative calculation of GW. We apply the method to a 1D U-V model with two sublattices carrying different chemical potentials. By comparing with those from DMRG, we show the results are shifted in the correct direction due to the GW contributions. {\bf 5)} In order to handle the generic Coulomb repulsion within E-DMFT, we describe a method to tailor E-DMFT so that proper momentum dependence can be kept in general response functions.

Conductivity of granular media with stagnant interstitial fluids via thermal particle dynamics simulation

In this paper, a numerical technique––the thermal particle dynamics method (TPD)––is extended to study heat conduction in granular media in the presence of stagnant interstitial fluids. The method, which generates a multi-particle simulation by explicitly modeling many two-particle interactions, allows bed heterogeneities to be directly included and dynamic temperature distributions to be obtained at the particle-level. Comparison with experimental data shows that TPD yields quantitatively accurate values of the effective thermal conductivity without introducing new adjustable parameters for a wide range of stagnant interstitial media. The model not only sheds light on fundamental issues in heat conduction in particulate materials, but also provides a valuable test bed for existing continuous theories.

Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix

Atomic and molecular ground-state energies are variationally determined by constraining the two-particle reduced density matrix (2-RDM) to satisfy positivity conditions. Because each positivity condition corresponds to correcting the ground-state energies for a class of Hamiltonians with two-particle interactions, these conditions collectively provide a new approach to many-body theory that, unlike perturbation theory, can capture significantly correlated phenomena including the multireference effects of potential-energy surfaces. The D, Q, and G conditions for the 2-RDM are extended through generalized lifting operators inspired from the formal solution of N-representability. These lifted conditions agree with the hierarchy of positivity conditions presented by Mazziotti and Erdahl [Phys. Rev. A 63, 042113 (2001)]. The connection between positivity and the formal solution explains how constraining higher RDMs to be positive semidefinite improves the N representability of the 2-RDM and suggests using pieces of higher positivity conditions that computationally scale like the D condition. With the D, Q, and G conditions as well as pieces of higher positivity the electronic energies for Be, LiH, ${\mathrm{H}}_{2}\mathrm{O},$ and BH are computed through a primal-dual interior-point algorithm for positive semidefinite programming. The variational method produces potential-energy surfaces that are highly accurate even far from the equilibrium geometry where single-reference perturbation-based methods often fail to produce realistic energies.

Influence of hydrodynamic interactions between particles on the turbulent flow in a suspension

A model based on the theory of Felderhof and Ooms [Phys. Fluids A 1 (1989) 1091] is used to evaluate the effect of hydrodynamic interactions between particles in turbulent two-phase flows. The particles are assumed to be point-particles, obeying Stokes' drag law. The flow of the suspending fluid phase is assumed to be (stochastic) stationary, homogeneous, isotropic turbulence. To simplify the calculations, only two-particle hydrodynamic interactions are taken into account. The value of the reduction function J, which is a measure of the hydrodynamic interactions, is evaluated as a function of the dimensionless wave number and frequency for a range of values of the density ratio between fluid and particles. Only for relatively heavy particles, hydrodynamic interactions affect the reduction function J and the corresponding spectral transfer functions. However, it was found that the effect of the interactions between point-particles could be neglected in turbulent flows, since very little energy is present at the frequencies where the interactions took place.

Thermophoretic interactions of aerosol particles with constant temperatures

This paper presents an analytical study of the thermophoretic motion of two free aerosol spheres with constant temperatures by using a method of reflections. The particles are allowed to differ in radius, in temperature, and in surface properties. The Knudsen numbers are assumed small so that a continuum model describes the fluid flow with a thermal creep and a hydrodynamic slip at the particle surfaces. The method of reflections is based on an analysis of the thermal and hydrodynamic disturbances produced by a single sphere with constant temperature placed in an arbitrarily varying temperature field. The results for two-sphere interactions are correct to O( r 12 −7), where r 12 is the distance between the particle centers. For the special situation of two identical spheres, the effect of particle interactions will drive the pair system approaching each other if the particle temperature is less than the temperature of the surrounding. While the temperature of the particles is higher than the surrounding temperature, the thermophoretic force obtains a repulsive effect between the particles. Based on a microscopic model the results for two-particle interactions are applied to find the effect of particle concentration on the average thermophoretic velocity in a bounded suspension. In general, the effect of interactions on thermophoretic coagulation of particles with constant temperatures can be stronger than that on sedimentation.

General Becker–Döring equations: effect of dimer interactions

The effect of dimer (two-particle) interactions on the Becker–Doring model of nucleation is investigated. Initially we consider the problem with size-independent aggregation and fragmentation coefficients and a constant monomer concentration. Either an equilibrium or a steady-state solution is found: the former when fragmentation is stronger than aggregation, the latter otherwise. By employing asymptotic techniques, the manner in which the system reaches these states is examined. The dimer interaction is found to accelerate the system towards the equilibrium solution, but has little impact on the relaxation time to the steady-state solution. In cases where aggregation is dominant, the steady-state cluster size distribution can only be determined consistently when the manner of approach to steady state is also known. In the terminology of asymptotic methods, one needs to know the first correction term in order to deduce the leading-order solution. We show how this can be derived and so at steady state we find a flux of matter to larger aggregation numbers due to monomer interactions, with a small and decreasing reverse flux due to dimer interactions. We then consider the case of constant density, that is allowing the monomer concentration to vary, and investigate the effect of a strong dimer interaction on the convergence to equilibrium. Two timescales are present and each one is investigated. We determine the intermediate meta-stable state, the final state and the timescales over which the system relaxes into these states. All results are shown to agree with numerical simulations.

Equation of state for helium-4 from microphysics

We compute the free energy of helium-4 near the lambda transition based on an exact renormalization-group equation. An approximate solution permits the determination of universal and nonuniversal thermodynamic properties starting from the microphysics of the two-particle interactions. The method does not suffer from infrared divergences. The critical chemical potential agrees with experiment. This supports a specific formulation of the functional integral that we have proposed recently. Our results for the equation of state reproduce the observed qualitative behavior. Despite certain quantitative shortcomings of our approximation, this demonstrates that ab initio calculations for collective phenomena become possible by modern renormalization-group methods.

Ensuring N-representability: Coleman's algorithm

The energy of a system which is described by a Hamiltonian which includes at most two-particle interactions can be expressed in terms of the second order reduced density matrix. However, for the 2-matrix to have proper symmetry is a weaker condition than requiring that the wavefunction be antisymmetric, which is called the N-representability problem, a problem of long term interest. Coleman [Reduced Density Matrices: Coulson's Challenge, Springer, New York, 2000] however, proposed an algorithm which ensures N-representability. In this Letter we examine the algorithm and show its connection to the full configuration interaction method and the contracted Schroedinger equation.

T-matrix analysis of biexcitonic correlations in the nonlinear optical response of semiconductor quantum wells

A detailed numerical analysis of exciton-exciton interactions in semiconductor quantum wells is presented. The theory is based on the dynamics-controlled truncation formalism and evaluated for the case of resonant excitation of 1s-heavy-hole excitons. It is formulated in terms of standard concepts of scattering theory, such as the forward-scattering amplitude (or T-matrix). The numerical diagonalization of the exciton-exciton interaction matrix in the 1s-approximation yields the excitonic T-matrix. We discuss the role of the direct and exchange interaction in the effective two-exciton Hamiltonian, which determines the T-matrix, evaluated within the 1s-subspace, and also analyze the effects of the excitonic wave function overlap matrix. Inclusion of the latter is shown to effectively prevent the 1s-approximation from making the Hamiltonian non-hermitian, but a critical discussion shows that other artefacts may be avoided by not including the overlap matrix. We also present a detailed analysis of the correspondence between the excitonic T-matrix in the 1s-approximation and the well-known T-matrix governing two-particle interactions in two dimensional systems via short-range potentials.

Deformed Chern–Simons interaction for nonrelativistic point particles

We deform the interaction between nonrelativistic point particles on a plane and a Chern-Simons field to obtain an action invariant with respect to time-dependent area-preserving diffeomorphisms. The deformed and undeformed Lagrangians are connected by a point transformation leading to a classical Seiberg-Witten map between the corresponding gauge fields. The Schroedinger equation derived by means of Moyal-Weyl quantization from the effective two-particle interaction exhibits - a singular metric, leading to a splitting of the plane into an interior (bag-) and an exterior region, - a singular potential (quantum correction) with singularities located at the origin and at the edge of the bag. We list some properties of the solutions of the radial Schroedinger equation.

Two charged particles in a two-dimensional well

Numerical solutions are obtained for the time-independent Schrödinger equation for a system of two particles of the same charge, repelled by the Coulomb force and confined to a two-dimensional infinite well. The eigenfunctions are expanded in a basis set of product delta functions; the Coulomb potential's singularity is thereby removed. We discuss our findings regarding correlated behavior in the lowest energy states of a well of width and length 3 bohr and the ordering of symmetric and antisymmetric states. Presented are conditional-probability densities in which Coulomb and Fermi holes are featured. The method is useful for treating other two-particle interactions in two dimensions. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Analytical results for trapped weakly interacting bosons in two dimensions

We consider a model of N two-dimensional bosons in a harmonic trap with translational and rotational invariant, weak two-particle interaction. We present in configuration space a systematical recursive method for constructing all wave functions with angular momentum L and corresponding energies and apply it to L\leq 6 for all N. The lower and the upper bounds for interaction energy are estimated. We analitically confirm the conjecture of Smith et al. that elementary symmetric polynomial is the ground state for repulsive delta interaction, for all N\geq L up to L\leq 6. Additionally, we find that there exist vanishing-energy solutions for L\geq N(N-1), signalizing the exclusive statistics. Finally, we consider briefly the case of attractive power-like potential r^k, k>-2, and prove that the lowest-energy state is still the one in which all angular momentum is absorbed by the center-of-mass motion.

Indirect interaction of Mössbauer nuclei

An expression is deduced for the operator of the indirect interaction of nuclei via the electromagnetic field. The properties of Mossbauer nuclei are described within the pseudospin formalism, which is usually used in the theory of optical two-level systems. The indirect interaction of pseudospins is derived by a method adopted from the theory of superconductivity. It is found that the potentials of this interaction involve terms decreasing as r−3, r−2, and r−1. The estimates demonstrate that the two-particle interaction can contribute significantly to the width of the resonance line, for example, in crystals whose cells contain thulium nuclei.

Perturbative spectrum of trapped weakly interacting bosons in two dimensions

We study a trapped Bose-Einstein condensate under rotation in the limit of weak, translational and rotational invariant two-particle interactions. We use the perturbation-theory approach (the large-N expansion) to calculate the ground-state energy and the excitation spectrum in the asymptotic limit where the total number of particles N goes to infinity while keeping the total angular momentum L finite. Calculating the probabilities of different configurations of angular momentum in the exact eigenstates gives us a clear view of the physical content of excitations. We briefly discuss the case of repulsive contact interaction.

Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles

For the description of ground-state correlation phenomena an accurate mapping of many-body quantum mechanics onto four particles is developed. The energy for a quantum system with no more than two-particle interactions may be expressed in terms of a two-particle reduced density matrix (2-RDM), but variational optimization of the 2-RDM requires that it corresponds to an N-particle wave function. We derive N-representability conditions on the 2-RDM that guarantee the validity of the uncertainty relations for all operators with two-particle interactions. One of these conditions is shown to be necessary and sufficient to make the RDM solutions of the dispersion condition equivalent to those from the contracted Schr\odinger equation (CSE) [Mazziotti, Phys. Rev. A 57, 4219 (1998)]. In general, the CSE is a stronger N-representability condition than the dispersion condition because the CSE implies the dispersion condition as well as additional N-representability constraints from the Hellmann-Feynman theorem. Energy minimization subject to the representability constraints is performed for a boson model with 10, 30, and 75 particles. Even when traditional wave-function methods fail at large perturbations, the present method yields correlation energies within 2%.

Multi-particle interaction in a model of the hydrophobic interaction

The multi-particle potential of mean force between interstitial solute molecules in Ben-Naim's one-dimensional, many-state lattice model is calculated. The solution is a direct extension of an earlier calculation of the two-particle interaction by Kolomeisky and Widom (Faraday Dis. 112 (1999) 81). It is found that the many-particle interaction potential is a sum of pair potentials between neighboring particles only. An exact equation of state, expressing the activity in the temperature and the pressure, is derived. The resulting solubility of a gaseous hydrophobe, which is defined osmotically, is calculated and found to increase considerably with the gas density.

Disorder and two-particle interaction in low-dimensional quantum systems

We review some of the recent results on two-interacting particles (TIP) in low-dimensional disordered quantum models. Special attention is given to the mapping of the problem onto random band matrices. In particular, we construct two simple, seemingly closely related examples for which an analogous mapping leads to incorrect results. We briefly discuss possible reasons for this discrepancy based on the physical differences between the TIP problem and our examples.

Parametric study of the 5d_2, 5d6s, and 5d6p configurations in the Yb I isoelectronic sequence

Data available on the 5d2,5d6s, and 5d6p configurations in the Yb i isoelectronic sequence have been critically reviewed by means of calculations with the orthogonal operators. The study included ions from Ta iv through Hg xi. Quite good agreement of the calculations with the experimental data for the ions from Ta iv through Au x has been achieved. A temporarily assigned value for the 5d2 1S0 level in the analysis of Re vi has been confirmed, and a previously unknown 5d6p 3P0 level in Au x has been established. For Hg xi, at least three of nine published levels in the 5d2 configuration and four of eleven levels in the 5d6p configuration are suspect because of their abnormal deviations from the calculated positions. The isoelectronic evolution of the parameter Amso that describes the two-particle magnetic ds interaction in the 5d6s configuration as well as of the two effective parameters S1 and S2 that describe higher-order effects in the 5d6p configuration was studied for the first time to the author’s knowledge in the Yb i sequence. The present calculations based on the orthogonal operators resulted in 5–10 times better agreement with the experimental data than did the conventional calculations.

Fermionic Renormalization Group Flows: Technique and Theory

We give a self-contained derivation of the differential equations for Wilson’s renormalization group for the one-particle irreducible Green functions in fermionic systems. The application of this equation to the (t, t � )-Hubbard model appears in Ref. 9). Here we focus on theoretical aspects. After deriving the equations, we discuss the restrictions imposed by symmetries on the effective action. We discuss scaling properties due to the geometry of the Fermi surface and give precise criteria to determine when they justify the use of one-loop flows. We also discuss the relationship of this approach to other RG treatments, as well as aspects of the practical treatment of truncated equations, such as the projection to the Fermi surface and the calculation of susceptibilities. Renormalization group (RG)studies of fermionic models are important for understanding models of solid-state theory and high-energy physics. In this paper, we set up a system of equations suitable for studying flows for general fermionic systems, in particular those with a Fermi surface at any density, and discuss general, but nontrivial, properties of their solutions. The system of equations applies both to “normal” and to symmetry-broken situations of fermionic systems with short-range interactions for d ≥ 1. By “short-range” we mean that the two-particle interaction decays so fast that its Fourier transform is a regular function. In Ref. 9)we have, together with Furukawa and Rice, applied the RG equations that we derive here to the two-dimensional (t, t � )-Hubbard model in the regime relevant for high-temperature superconductivity. The purpose of the present paper is to give more background on the theoretical aspects of the equations, in particular on a detailed argument which justifies the one-loop flow in a certain scale range, even if the scale-dependent interaction is no longer small. In the following, we give an overview; the details are filled in in the later sections. 1.1. The RG and the three scale regimes The Wilsonian RG organizes the functional integration corresponding to the grand canonical trace as an iterated integral over degrees of freedom with energies in a certain range, i.e. those that belong to a corresponding length scale. As such, it is simply a rewriting of the generating functional of the correlation functions as a function of an energy scale � . In contrast to the situation in other RG schemes, this approach does not require any assumptions about perfect scaling laws as a

Dephasing kinetics of a coherent exciton polarization by exciton–exciton scattering

A system of excitons interacting via a two-particle interaction and with an external light field is treated within a boson approximation. Equations are derived and solved numerically including both the coherent interaction of the excitons and the incoherent x–x scattering on a microscopic level. Anomalous propagators, consisting of two exciton annihilation operators, have been also included. These are found to be necessary for the generation of incoherent excitons after coherent resonant excitation. The complete excitation, thermalization and dephasing kinetics can be studied within our model. A considerable slowing down of the dephasing is found for exciton densities approaching the critical conditions for a Bose–Einstein condensation showing strong deviations from a simple linear excitation-induced dephasing. This can be interpreted as a first sign of the onset of this phase transition.