Single-particle Density Research Articles

Quantum transport and the Wigner distribution function for Bloch electrons in spatially homogeneous electric and magnetic fields

The theory of Bloch electron dynamics for carriers in homogeneous electric and magnetic fields of arbitrary time dependence is developed in the framework of the Liouville equation. The Wigner distribution function (WDF) is determined from the single particle density matrix in the ballistic regime, i.e., collision effects are excluded. The single particle transport equation is established with the electric field described in the vector potential gauge, and the magnetic field is treated in the symmetric gauge. The general approach is to employ the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including multiband Zener tunneling, is treated exactly. In the formulation of the WDF, we transform to a new set of variables so that the final WDF is gauge invariant and is expressed explicitly in terms of the position, kinetic momentum, and time. The methodology for developing the WDF is illustrated by deriving the exact WDF equation for free electrons in homogeneous electric and magnetic fields. The methodology is then extended to the case of electrons described by an effective Hamiltonian corresponding to an arbitrary energy band function. In treating the problem of Bloch electrons in a periodic potential, the methodology for deriving the WDF reveals a multiband character due to the inherent nature of the Bloch states. In examining the single-band WDF, it is found that the collisionless WDF equation matches the equivalent Boltzmann transport equation to first order in the magnetic field. These results are necessarily extended to second order in the magnetic field by employing a unitary transformation that diagonalizes the Hamiltonian using the ABR to second order. The work includes a discussion of the multiband WDF transport analysis and the identification of the combined Zener-magnetic field induced tunneling.

A novel method for determining the single particle potential directly from the measured single particle density: Application to the charge density difference between the isotones 206Pb–205Tl

We present a novel method, based on the single particle Schroedinger equation, to determine the central potential (mean-field) directly from the single particle matter density and its first and second derivatives. As an example, we consider the experimental data for the charge density difference between the isotones 206Pb–205Tl, deduced by phase shift analysis of elastic electron scattering cross-section measurements and corresponds to the shell model 3s1/2 proton orbit, and determine the corresponding single particle potential. We also present results of least-square fits to parametrized single particle potentials. The 3s1/2 wave functions of the determined potentials reproduce fairly well the experimental data within the quoted errors. More accurate experimental data, with uncertainty smaller by a factor of two or more, may answer the question how well can the data be reproduced by a calculated 3s1/2 wave function.

Single-cone finite difference scheme for the (2+1)D Dirac von Neumann equation

An explicit finite difference scheme is presented for the von Neumann equation for (2+1)D Dirac fermions. It is founded upon a staggered space–time grid which ensures a single-cone energy dispersion and performs the time-derivative in one sweep using a three-step leap-frog procedure. It enables a space–time-resolved numerical treatment of the mixed-state dynamics of Dirac fermions within the effective single-particle density matrix formalism. Energy-momentum dispersion, stability and convergence properties are derived. Elementary numerical tests to demonstrate stability properties use parameters which pertain to topological insulator surface states. A method for the simulation of charge injection from an electric contact is presented and tested numerically. Potential extensions of the scheme to a Dirac–Lindblad equation, real-space–time Green's function formulations, and higher-order finite-difference schemes are discussed.

Simple correlated wave-function for excitons in 0D, quasi-1D and quasi-2D quantum dots

We propose correlated yet extremely simple single-parameter-dependent wave-function with a Slater-type correlation factor, to describe excitons in 0D, quasi-1D and quasi-2D semiconductor quantum dots. We provide closed-form formulas for the wave-function normalization factor, electron/hole single-particle density and the expectation value of the kinetic energy. We additionally supply fast integration procedures for the Coulomb interaction in the presence of dielectric mismatch with the surrounding medium for nanoplatelets (quasi-2D systems), and for the bare Coulomb integral in long nanorods (quasi-1D systems).

Single-photon double ionization: renormalized-natural-orbital theory versus multi-configurational Hartree–Fock

The N-particle wavefunction has too many dimensions for a direct time propagation of a many-body system according to the time-dependent Schrödinger equation (TDSE). On the other hand, time-dependent density functional theory (TDDFT) tells us that the single-particle density is, in principle, sufficient. However, a practicable equation of motion for the accurate time evolution of the single-particle density is unknown. It is thus an obvious idea to propagate a quantity which is not as reduced as the single-particle density but less dimensional than the N-body wavefunction. Recently, we have introduced time-dependent renormalized-natural-orbital theory (TDRNOT). TDRNOT is based on the propagation of the eigenfunctions of the one-body reduced density matrix, the so-called natural orbitals. In this paper we demonstrate how TDRNOT is related to the multi-configurational time-dependent Hartree–Fock (MCTDHF) approach. We also compare the performance of MCTDHF and TDRNOT versus the TDSE for single-photon double ionization (SPDI) of a 1D helium model atom. SPDI is one of the effects where TDDFT does not work in practice, especially if one is interested in correlated photoelectron spectra, for which no explicit density functional is known.

On the ground state energy of the inhomogeneous Bose gas

Within the self-consistent Hartree–Fock approximation, an explicit in this approximation expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose–Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose–Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross–Pitaevski approach, is found. The obtained form of kinetic energy is valid beyond the Hartree–Fock approximation and can be applied for arbitrary strong interparticle interaction.

Single-Particle Excitations and Effects of Hetero-Pairing Fluctuations in a Bose–Fermi Mixture with a Feshbach Resonance

We theoretically investigate normal-state properties of a gas mixture of single-component bosons and fermions with a hetero-nuclear Feshbach resonance. Including strong hetero-pairing fluctuations associated with the Feshbach resonance, we calculate single-particle density of states, as well as the spectral weight at various interaction strengths. For this purpose, we employ an improved T-matrix approximation (TMA), where the bare Bose Green's function in the non-selfconsistent TMA self-energy is modified so as to satisfy the Hugenholtz-Pines relation at the Bose-Einstein condensation (BEC) temperature $T_{\rm BEC}$. In the unitary regime at $T_{\rm BEC}$, we show that hetero-pairing fluctuations couple Fermi atomic excitations with Fermi molecular excitations, as well as with Bose atomic excitations. Although a similar coupling phenomenon by pairing fluctuations is known to give a pseudo-gapped density of states in the unitary regime of a two-component Fermi gas, such a dip structure is found to not appear even in the unitary limit of a Bose-Fermi mixture. It only appears in the strong-coupling regime. Instead, a spectral peak along the molecular dispersion appears in the spectral weight.We also clarify how this coupling phenomenon is seen in the Bose channel. Since a hetero-nuclear Feshbach resonance, as well as the formation of Bose-Fermi molecules, have been realized, our results would be useful for the study of strong-coupling properties of this unique quantum gas.

Lindblad approach to spatiotemporal quantum dynamics of phonon-induced carrier capture processes

In view of the ultrashort spatial and temporal scales involved, carrier capture processes in nanostructures are genuine quantum phenomena. To describe such processes, methods with different levels of approximations have been developed. By properly tailoring the Lindblad-based nonlinear single-particle density matrix equation provided by an alternative Markov approach, in this work we present a Lindblad superoperator to describe how the phonon-induced carrier capture affects the spatio-temporal quantum dynamics of a flying wave packet impinging on a quantum dot. We compare the results with non-Markovian quantum kinetics calculations and discuss the advantages and drawbacks of the two approaches.

Spin-orbit-coupled atomic Fermi gases in two-dimensional optical lattices in the presence of a Zeeman field

We investigate the single-particle and collective excitations of a Rashba spin-orbit coupled atomic Fermi gas with attractive interaction, loaded in a two dimensional (2D) square optical lattice, in the presence of an effective out-of-plane Zeeman field. Our numerical calculations show that the many body physics of the Bardeen-Cooper-Schrieffer (BCS) side is strongly modified compared to the Fermi gases in the free space. The physics behind this statement is in the fact that without a lattice structure, if the value of the Zeeman field does not exceed some threshold value, the minimum of the single-particle ground state energy is infinitely degenerate and occurs along a ring in $(k_x,k_y)$ space. This reduces the effective dimensionality; the single-particle density of states is a constant at low energies, and the molecular pairing is strongly enhanced. In contrast to the continuum, in the 2D lattice case there are only four degenerate minima, and because of that, the spin-orbit coupling (SOC) in optical lattices gives rise to unusual properties entirely different from the continuum. For example, in the continuum, the pairing gap as well as the condensate fraction are strongly enhanced by the SOC strength on the BCS side. In the presence of lattice geometry the gap and the condensate fraction increase as a function of the SOC strength only at the small fillings and weak attraction limit. Moreover, we found that the speed of sound also exhibits different behavior: in the 3D and 2D continuum, the slope of the Goldstone mode decreases as a function of the SOC strength, while in the lattice case the speed of sound increases monotonically with SOC strength.

The single-particle density matrix of a quantum bright soliton from the coordinate Bethe ansatz

We present a novel approach for computing reduced density matrices for superpositions of eigenstates of a Bethe-ansatz solvable model by direct integration of the wave function in coordinate representation. A diagrammatic approach is developed to keep track of relevant terms and identify symmetries, which helps to reduce the number of terms that have to be evaluated numerically. As a first application we compute with modest numerical resources the single-particle density matrix and its eigenvalues including the condensate fraction for a quantum bright soliton with up to N = 10 bosons. The latter are constructed as superpositions of string-type Bethe-ansatz eigenstates of nonrelativistic bosons in one spatial dimension with attractive contact interaction. Upon delocalising the superposition in momentum space we find that the condensate fraction reaches maximum values larger than 97% with weak particle-number dependence in the range of particles studied. The presented approach is suitable for studying time-dependent problems and generalises to higher-order correlation functions.

The single-particle characteristics of Pb isotopes near the drip lines calculated within the dispersive optical model

The neutron and proton dispersive optical potential for the 208Pb nucleus has been determined for the energy region from–70 to +60 MeV and used to calculate the differential elastic scattering, the total interaction and reaction cross sections, as well as the single-particle characteristics, the neutron and charge densities, rms radii, and the thickness of the nucleus skin. The calculated results are in good agreement with the experimental data. The proton dispersive optical model potential for the spherical and close to spherical Pb isotopes within the neutron and proton drip lines has been obtained by a similar method. The calculation predicts a trend towards the growth of the proton particle-hole gap, which corresponds to Z = 82 shell closure as Z approaches the proton drip line.

Nuclear Energy Density Functional for KIDS

The density functional theory (DFT) is based on the existence and uniqueness of a universal functional $E[\rho]$, which determines the dependence of the total energy on single-particle density distributions. However, DFT says nothing about the form of the functional. Our strategy is to first look at what we know, from independent considerations, about the analytical density dependence of the energy of nuclear matter and then, for practical applications, to obtain an appropriate density-dependent effective interaction by reverse engineering. In a previous work on homogeneous matter, we identified the most essential terms to include in our "KIDS" functional, named after the early-stage participating institutes. We now present first results for finite nuclei, namely the energies and radii of $^{16,28}$O, $^{40,60}$Ca.

Hierarchical equations of motion approach to transport through an Anderson impurity coupled to interacting Luttinger liquid leads

We generalize the hierarchical equations of motion method to study electron transport through a quantum dot or molecule coupled to one-dimensional interacting leads that can be described as Luttinger liquids. Such leads can be realized, for example, by quantum wires or fractional quantum Hall edge states. In comparison to noninteracting metallic leads, Luttinger liquid leads involve many-body correlations and the single-particle tunneling density of states shows a power-law singularity at the chemical potential. Using the generalized hierarchical equations of motion method, we assess the importance of the singularity and the next-to-leading order many-body correlations. To this end, we compare numerically converged results with second and first-order results of the hybridization expansion that is inherent to our method. As a test case, we study transport through a single-level quantum dot or molecule that can be described by an Anderson impurity model. Cotunneling effects turn out to be most pronounced for attractive interactions in the leads or repulsive ones if an excitonic coupling between the dot and the leads is realized. We also find that an interaction-induced negative differential conductance near the Coulomb blockade thresholds is slightly suppressed as compared to a first-order and/or rate equation result. Moreover, we find that the two-particle ($n$-particle) correlations enter as a second-order ($n$-order) effect and are, thus, not very pronounced at the high temperatures and parameters that we consider.

Influence of nuclear deformation on level density parameter

By using semiclassical method and considering deformed Woods–Saxon mean field potential, single-particle level density g, level density parameter a, and nuclear level density have been calculated for some deformed nuclei with deformation parameters ([Formula: see text], [Formula: see text]). Significant difference between deformed and spherical level density parameter and nuclear level density were observed. The effect of [Formula: see text] on level density parameter was notable. A simple formula for single-particle level density has been introduced.

Quantum Embedding Theories.

In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations-a high level calculation on the active region of interest, and a low level calculation on its environment-formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: density functional embedding, Green's function embedding, and density matrix embedding, respectively, use the single-particle density, single-particle Green's function, and single-particle density matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for density functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for density matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theoretical perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with density functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field density functional embedding and wave function in density functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the density functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in density functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed. The third section concerns density matrix embedding. Here, we first highlight some mathematical complications associated with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the density matrix embedding theory, where we use the Schmidt decomposition to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency associated with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in density functional embedding. We finish with perspectives on the future of all three methods.

On the Ground-State Energy and Local Pressure of an Inhomogeneous Bose Gas

Within the self-consistent Hartree-Fock approximation, an explicit expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose-Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose-Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross-Pitaevski approach, is found. This form is valid beyond the Hartree-Fock approximation and can be applied for arbitrary strong interparticle interaction.

Andreev reflection and bound state formation in a ballistic two-dimensional electron gas probed by a quantum point contact

We study coherent transport and bound-state formation of Bogoliubov quasiparticles in a high-mobility In$_{0.75}%$Ga$_{0.25}$As two-dimensional electron gas (2DEG) coupled to a superconducting Nb electrode by means of a quantum point contact (QPC) as a tunable single-mode probe. Below the superconducting critical temperature of Nb, the QPC shows a single-channel conductance greater than the conductance quantum $2e^{2}/h$ at zero bias, which indicates the presence of Andreev-reflected quasiparticles, time-reversed states of the injected electron, returning back through the QPC. The marked sensitivity of the conductance enhancement to voltage bias and perpendicular magnetic field suggests a mechanism analogous to reflectionless tunneling--a hallmark of phase-coherent transport, with the boundary of the 2DEG cavity playing the role of scatters. When the QPC transmission is reduced to the tunneling regime, the differential conductance vs bias voltage probes the single-particle density of states in the proximity area. Measured conductance spectra show a double peak within the superconducting gap of Nb, demonstrating the formation of Andreev bound states in the 2DEG. Both of these results, obtained in the open and closed geometries, underpin the coherent nature of quasiparticles, i.e., phase-coherent Andreev reflection at the InGaAs/Nb interface and coherent propagation in the ballistic 2DEG.

Study of weakly interacting trapped Bose gas

We have obtained expressions for single particle density and two particle density of weakly interacting trapped quantum gases. These are valid for all temperature and in any dimension. These expressions have been simplified and expressed in terms of non-interacting single particle density. The ground fluctuations for T<Tc in grand canonical ensemble has been treated with care using the method of Kocharovsky et al. [Phys. Rev. A 61, 053606 (2000)]. Some numerical results are presented in one and three dimension for isotropic harmonically trapped Bose gas with contact interactions. It is seen that boson density decreases with increasing repulsive interactions. The expression for critical temperature is also shown to agree with earlier result and is in accordance with experiments.

Spin susceptibility of disordered gapped graphene systems

We calculate the spin susceptibility for the case of gapped graphene systems in the presence of disorder. The average single-particle density of states in gapped graphene with disorder was calculated, using the Born and the T-matrix approximations. The temperature dependence of the static spin susceptibility was analyzed. The influence of the chemical potential position and disorder is also discussed.

Properties of bosons in a one-dimensional bichromatic optical lattice in the regime of the pinning transition: A worm-algorithm Monte Carlo study

The properties of interacting bosons in a weak, one-dimensional, and bichromatic optical with a rational ratio of the constituting wavelengths $\lambda_1$ and $\lambda_2$ are numerically examined along a broad range of the Lieb-Liniger interaction parameter $\gamma$ passing through the Sine-Gordon transition. It is argued that there should not be much difference in the results between those due to an irrational ratio $\lambda_1/\lambda_2$ and due to a rational approximation of the latter. For a weak bichromatic optical lattice, it is chiefly demonstrated that this transition is robust against the introduction of quasidisorder via a weaker, secondary, and incommensurate optical lattice superimposed on the primary one. The properties, such as the correlation function, Matsubara Green's function, and the single-particle density matrix, do not respond to changes in the depth of the secondary optical lattice $V_1$. For a stronger bichromatic optical lattice, however, a response is observed because of changes in $V_1$. It is found accordingly, that holes in the SG regime play an important role in the response of properties to changes in $\gamma$. The continuous-space worm algorithm Monte Carlo method [Boninsegni \ea, Phys. Rev. E $\mathbf{74}$, 036701 (2006)] is applied for the present examination. It is found that the worm algorithm is able to reproduce the Sine-Gordon transition that has been observed experimentally [Haller \ea, Nature $\mathbf{466}$, 597 (2010)].