Two-particle Problem Research Articles

Repulsive bound-electron pairs in a Peierls lattice

A new class of repulsive bound-electron pairs have been found in a Peierls lattice within the Hubbard model for energies E < 0 despite U > 0. These new repulsive bound-electron pairs have a high degree of localization as both the correlation energy U and the tS/tL hopping ratio increases. In order to study electronic correlation in Peierls lattices, our previous real space mapping method has also been extended to the so called generalized mapping method which is briefly presented here. In this paper, we concentrate our attention to discuss in detail the two-particle problem within a repulsive Hubbard model for the one-dimensional Peierls lattice.

Relationship between two-particle topology and fractional Chern insulator

Lattice generalizations of fractional quantum Hall (FQH) systems, called fractional Chern insulators (FCIs), have been extensively investigated in strongly correlated systems. Despite many efforts, previous studies have not revealed all of the guiding principles for the FCI search. In this paper, we investigate a relationship between the topological band structure in the two-particle problem and the FCI ground states in the many-body problem. We first formulate the two-particle problem of a bosonic on-site interaction projected onto the lowest band of a given tight-binding Hamiltonian. We introduce a reduced Hamiltonian whose eigenvalues correspond to the two-particle bound-state energies. By using the reduced Hamiltonian, we define the two-particle Chern number and numerically check the bulk-boundary correspondence that is predicted by the two-particle Chern number. We then propose that a nontrivial two-particle Chern number of dominant bands roughly indicates the presence of bosonic FCI ground states at filling factor $\ensuremath{\nu}=1/2$. We numerically investigate this relationship in several tight-binding models with Chern bands and find that it holds well in most of the cases, albeit two-band models being exceptions. Although the two-particle topology is neither a necessary nor a sufficient condition for the FCI state as other indicators in previous studies, our numerical results indicate that the two-particle topology characterizes the degree of similarity to the FQH systems.

One- and two-particle problem with correlated disorder potential

Motivated by the recent experimental and theoretical progresses in the exploration of the effect of disorder in interacting system, we examine the effect of two types of correlated disorder, the quasi-periodic potential and speckle disorder potential, on one- and two-particle problem with exact diagonalization (ED) method. We give the phase diagram for single particle in the presence of quasi-periodic potential and also analyse the effect of strong interaction on the phase diagram for ground state in two dimensions. For the speckle disorder potential case, we examine both the effect of correlation length and disorder strength on single particle ground state energy and two-particle binding energy. The transport property for different interaction strength under speckle disorder potential is also calculated and discussed at last.

Hydrodynamic dispersion and aggregation induced by shear in non-Brownian magnetic suspensions

In this paper, we examine a two-particle problem in order to study transport phenomena on magnetic suspensions such as the shear-induced hydrodynamic diffusion and the shear-induced aggregation in the regime of non-Brownian particles and creeping flow. New results are presented for the shear-induced hydrodynamic diffusivities and the rate of particle doublet formation, resulting from the diffusive and aggregative irreversible trajectories produced by particle magnetic interactions. The numerical computation of the particle shear-induced diffusivities and the rates of particle aggregation are performed by using a Monte Carlo integration scheme for different values of the relevant magnetic parameter α. This parameter represents the non-dimensional strength of the dipole-dipole magnetic interactions between the particles. For small values of α, the shear-induced self-diffusivity is remarkably described by a slight adaptation of the asymptotic theory for rough spherical particles interacting hydrodynamically [F. R. Cunha and E. J. Hinch, “Shear-induced dispersion in a dilute suspension of rough spheres,” J. Fluid Mech. 309, 211 (1996)]. We just replace the roughness parameter ϵ with α5/4, giving for the self-shear-induced diffusivity for small values of α, 0.156γ̇a2ϕα0.547log(1/α5/4)+1.347−0.701, where γ̇ is the applied shear rate, a is the radius of the magnetic spheres, and ϕ is the particle volume fraction. In addition, a root-square law dependence is obtained for the rate of particle aggregation as (1.830/π)ϕ1N1γ̇α1/2, where N1 and ϕ1 are, respectively, the number and the volume fraction of the isolated particles in the suspension. A comparison shows that the root-square law prediction is in excellent agreement with the results of the numerical simulations for all values of the parameter α investigated.

Numerical investigation of the dynamics of Janus magnetic particles in a rotating magnetic field

We investigated the rotational dynamics of Janus magnetic particles suspended in a viscous liquid, in the presence of an externally applied rotating magnetic field. A previously developed two-dimensional direct simulation method, based on the finite element method and a fictitious domain method, is employed to solve the magnetic particulate flow. As for the magnetic problem, the two Maxwell equations are converted to a differential equation using the magnetic potential. The magnetic forces acting on the particles are treated by a Maxwell stress tensor formulation, enabling us to consider the magnetic interactions among the particles without any approximation. The dynamics of a single particle in the rotating field is studied to elucidate the effect of the Mason number and the magnetic susceptibility on the particle motions. Then, we extended our interest to a two-particle problem, focusing on the effect of the initial configuration of the particles on the particle motions. In three-particle interaction problems, the particle dynamics and the fluid flow induced by the particle motions are significantly affected by the particle configuration and the orientation of each particle.

Two-particle problem in comblike structures

Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comblike structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comblike architecture can strongly differ according to whether both reactants are moving (no matter their relative diffusivities) or only one is moving and according to the density of shortcuts among the branches.

Anderson localization of Cooper pairs and Majorana fermions in an ultracold atomic Fermi gas with synthetic spin-orbit coupling

We theoretically investigate two-particle and many-particle Anderson localizations of a spin-orbit coupled ultracold atomic Fermi gas trapped in a quasi-periodic potential and subjected to an out-of-plane Zeeman field. We solve exactly the two-particle problem in a finite length system by exact diagonalization and solve approximately the many-particle problem within the mean-field Bogoliubov-de Gennes approach. At a small Zeeman field, the localization properties of the system are similar to that of a Fermi gas with conventional $s$-wave interactions. As the disorder strength increases, the two-particle binding energy increases and the fermionic superfluidity of many particles disappears above a threshold. At a large Zeeman field, where the interatomic interaction behaves effectively like a $p$-wave interaction, the binding energy decreases with increasing disorder strength and the resulting topological superfluidity shows a much more robust stability against disorder than the conventional $s$-wave superfluidity. We also analyze the localization properties of the emergent Majorana fermions in the topological phase. Our results could be experimentally examined in future cold-atom experiments, where the spin-orbit coupling can be induced artificially by using two Raman lasers, and the quasi-periodic potential can be created by using bichromatic optical lattices.

Reversible Stochastically Gated Diffusion-Influenced Reactions.

An approximate but accurate theory is developed for the kinetics of reversible binding of a ligand to a macromolecule when either can stochastically fluctuate between reactive and unreactive conformations. The theory is based on a set of reaction-diffusion equations for the deviations of the pair distributions from their bulk values. The concentrations are shown to satisfy non-Markovian rate equations with memory kernels that are obtained by solving an irreversible geminate (i.e., two-particle) problem. The relaxation to equilibrium is not exponential but rather a power law. In the Markovian limit, the theory reduces to a set of ordinary rate equations with renormalized rate constants.

Magnetic interaction of Janus magnetic particles suspended in a viscous fluid.

We studied the magnetic interaction between circular Janus magnetic particles suspended in a Newtonian fluid under the influence of an externally applied uniform magnetic field. The particles are equally compartmentalized into paramagnetic and nonmagnetic sides. A direct numerical scheme is employed to solve the magnetic particulate flow in the Stokes flow regime. Upon applying the magnetic field, contrary to isotropic paramagnetic particles, a single Janus particle can rotate due to the magnetic torque created by the magnetic anisotropy of the particle. In a two-particle problem, the orientation of each particle is found to be an additional factor that affects the critical angle separating the nature of magnetic interaction. Using multiparticle problems, we show that the orientation of the particles has a significant influence on the dynamics of the particles, the fluid flow induced by the actuated particles, and the final conformation of the particles. Straight and staggered chain structures observed experimentally can be reproduced numerically in a multiple particle problem.

Micromechanics prediction of effective modulus for asphalt mastic considering inter-particle interaction

This paper presents the development and validation of a new micromechanical model, Ju–Chen (J–C) model, to predict the effective viscoelastic modulus of asphalt mastic by considering the inter-particle interaction. Based on the approximate solutions for two-particle interaction problem, the radial distribution function is integrated into the ensemble-volume averaged eigenstrain tensor so as to consider the inter-particle interaction, and the solution could be extended and simplified to predict the effective complex modulus of asphalt mastic with different filler volume fractions. It is found that the inter-particle interaction increases with the filler volume fraction and decreases with the test frequency, and the predictions agree well with the experiment data at low and moderate filler volume fractions, not exceeding 50%. Compared with two commonly used micromechanical models (M–T model and DSEM model), the J–C model relatively gives the best estimation of the effective modulus of mastics, and the applicability of J–C model could be further improved.

Dimensionality effect on two-electron energy spectrum: A fractional-dimension-based formulation

We analyze the spectrum of two-electron quantum dot with anisotropic parabolic confinement by using the fractional-dimension formulation which allows us to reduce the two-particle problem to two equations for independent particles in an effective space with variable dimension, ranging between two and three for ellipsoidal-shaped quantum dots and between one and two for elliptical-shaped quantum disks. The dependencies of energy levels on the heterostructure sizes for quantum dots, disks and wires are presented.

Slow encounters of particle pairs in branched structures.

On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent; i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold, and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comblike structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we provide evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one-particle problem.

Hierarchy of Fractional Chern Insulators and Competing Compressible States

We study the phase diagram of interacting electrons in a dispersionless Chern band as a function of their filling. We find hierarchy multiplets of incompressible states at fillings ν = 1/3, 2/5, 3/7, 4/9, 5/9, 4/7, 3/5 as well as ν = 1/5, 2/7. These are accounted for by an analogy to Haldane pseudopotentials extracted from an analysis of the two-particle problem. Important distinctions to standard fractional quantum Hall physics are striking: in the absence of particle-hole symmetry in a single band, an interaction-induced single-hole dispersion appears, which perturbs and eventually destabilizes incompressible states as ν increases. For this reason, the nature of the state at ν = 2/3 is hard to pin down, while ν = 5/7, 4/5 do not seem to be incompressible in our system.

Direct simulation of the dynamics of two spherical particles actuated magnetically in a viscous fluid

A three-dimensional direct simulation method is developed to describe flows with paramagnetic particles suspended in a non-magnetic fluid. We validate the method by comparing particle motions obtained from our scheme with those from a particle dynamics (PD) method and a theoretical model based on the point-dipole approximation, using a two-particle problem in a uniform field. The critical angle separating the nature of magnetic interaction is found to be affected by the distance between the two particles. When hydrodynamic interactions between particles are not taken into account, trajectories obtained from the PD method show severe deviations from those computed using the direct simulation method even in the simple two-particle interaction problem. The dynamics of a two-particle chain in a rotating field are highly influenced by the Mason number, which is the ratio of viscous force to magnetic force. Two regimes of the Mason number are found, showing two distinctive behaviors: (i) rigid body rotation with a phase lag and (ii) oscillatory motion on top of global rotation. The critical Mason number and the dimensionless period of oscillatory motion are affected by the magnetic susceptibility of the particles.

Entanglement dynamics in a non-Markovian environment: An exactly solvable model

We study the non-Markovian effects on the dynamics of entanglement in an exactly-solvable model that involves two independent oscillators each coupled to its own stochastic noise source. First, we develop Lie algebraic and functional integral methods to find an exact solution to the single-oscillator problem which includes an analytic expression for the density matrix and the complete statistics, i.e., the probability distribution functions for observables. For long bath time-correlations, we see non-monotonic evolution of the uncertainties in observables. Further, we extend this exact solution to the two-particle problem and find the dynamics of entanglement in a subspace. We find the phenomena of `sudden death' and `rebirth' of entanglement. Interestingly, all memory effects enter via the functional form of the energy and hence the time of death and rebirth is controlled by the amount of noisy energy added into each oscillator. If this energy increases above (decreases below) a threshold, we obtain sudden death (rebirth) of entanglement.

Low-density molecular gas of tightly bound Rashba-Dresselhaus fermions

We study interacting Rashba-Dresselhaus fermions in two spatial dimensions. First, we present an exact solution to the two-particle bound-state problem of spin-orbit (SO)-coupled fermions for arbitrary Rashba and Dresselhaus SO interactions. An exact molecular-wave function and the Green's function are derived explicitly along with the binding energy and the spectrum of the molecular state. In the second part, we consider a thermal Boltzmann gas of fermionic molecules and compute the time-of-flight (TOF) velocity and spin distributions for a single fermion in the gas. We show that the signatures of the bound state can be observed already in the first-moment expectation values, such as TOF density and spin profiles.

Self-propulsion and dispersion of reactive colloids due to entropic anisotropy

In this paper, self-motion of reactive colloids and their dispersion behaviour are theoretically examined. The motion is driven by an osmotic force imbalance arising from non-uniform atmospheres of reactive solutes around the colloids. The propulsion here is not limited to Janus-like particles. It can also occur to particles having ‘uniform’ reactivity due to the more universal mechanism – entropic anisotropy created by breaking in rotational symmetry. The idea is demonstrated by examining the motion of a reactive particle due to asymmetry in its shape or to the presence of an additional particle. In the two-particle problem, in particular, we find that sink (source) particles can self-migrate towards (apart from) each other at velocities varying asR−2, resembling Coulomb attraction (repulsion), whereRis the inter-particle distance. Because of this Coulomb-like nature, a suspension of sink particles could undergo collective flocculation due to unscreened osmotic attraction. The criterion for an occurrence of the flocculation is also established. It reveals that the flocculation can occur if the particle volume fraction is within a certain window in terms of the solute concentration and the particle reactivity. The stability of reactive suspensions is also discussed using the modified Derjaguin–Landau–Verwey–Overbeek (DLVO) theory that takes account of the competition between long-range reaction-induced osmotic forces and short-range colloidal forces. A more generalized view for the present self-driven particle motion is elucidated by a simple scaling theory, providing lucid accounts for the self-motion of two particles, composite bodies, and Janus particles – all are driven by dipolar distortions in potential energy. Comparison with phoretic self-swimmers is also discussed.

An optimized description of a confined interacting Fermi system

We provide an efficient form to express the action of a many-body Hamiltonian of harmonically trapped interacting Fermi particles on wavefunctions built from paired states. The expression is suitable to numerically determine the ground state energy, regardless of the form of the two-body interaction. It takes advantage of the knowledge of the two-particle problem and the inherent properties of the matrix form of the many-body wavefunction. As an example, we evaluate the properties of a system composed of a balanced mixture of two families of fermions confined in a harmonic trap interacting through a short-range exponential potential. Numerical results for N ⩽ 10 and N = 35, 56, 84 and 165 particles of each family are reported. In the strong interacting regime corresponding to an infinite s-wave scattering length, our results give an upper bound to the Bertsch parameter for harmonically trapped systems (E/EIFG)2 = 1 + β ⩽ 0.376 ± 0.008 with E the total energy and EIFG the energy for the analogous ideal Fermi gas. The influence of the harmonic trap and the interaction potential is exhibited in one and two-body correlation functions.

Interaction between like-charged particles at a liquid interface: Electrostatic repulsion vs. electrocapillary attraction

The electric field of charged particles, which are adsorbed at a liquid interface, induces interfacial deformations (capillary menisci). The overlap of such deformations gives rise to electrocapillary force of interaction between the particles. Our goal is to quantify this interaction on the basis of a force approach, which is different from the approaches (mostly based on energy calculations) used by other authors. The fact that the electric field of adsorbed particles has a dipolar asymptotics (due to the image-charge effect) is utilized to derive an analytical expression for the meniscus profile. The comparison of the calculated profile with experimental data indicates that the results based on the dipolar approximation agree excellently with the data, except some small deviations near the contact line. The effect of the interfacial deformation on the electrostatic pressure is also taken into account. The two-particle electrocapillary problem is solved in bipolar coordinates without using the superposition approximation. It turns out that for uniform distribution of the surface charges, the electrocapillary attraction is weaker than the electrostatic repulsion at interparticle distances at which the dipolar approximation is applicable, so that the net force is repulsive. This result is in agreement with the conclusions of other authors obtained by using different theoretical approaches and with available experimental data. The analytical expressions for the electrocapillary and electrodipping forces derived in the present article provide a simple and convenient way for estimation of these forces.

Electronic structure and optical properties of quantum-confined lead salt nanowires

In the framework of four-band envelope-function formalism, developed earlier for spherical semiconductor nanocrystals, we study the electronic structure and optical properties of quantum-confined lead-salt (PbSe and PbS) nanowires (NWs) with a strong coupling between the conduction and the valence bands. We derive spatial quantization equations, and calculate numerically energy levels of spatially quantized states of a transverse electron motion in the plane perpendicular to the NW axis, and electronic subbands developed due to a free longitudinal motion along the NW axis. Using explicit expressions for eigenfunctions of the electronic states, we also derive analytical expressions for matrix elements of optical transitions and study selection rules for interband absorption. Next we study a two-particle problem with a conventional long-range Coulomb interaction and an interparticle coupling via medium polarization. The obtained results show that due to a large magnitude of the high-frequency dielectric permittivity of PbSe material, and hence, a high dielectric NW/vacuum contrast, the effective coupling via medium polarization significantly exceeds the effective direct Coulomb coupling at all interparticle separations along the NW axis. Furthermore, the strong coupling via medium polarization results in a bound state of the longitudinal motion of the lowest-energy electron-hole pair (a longitudinal exciton), while fast transverse motions of charge carriers remain independent of each other.