Abstract. In this paper, a gradient flow model is presented for conducting ground state calculations in the Wigner formalism of many-body systems in the framework of density functional theory. Theoretically, an energy functional in the Wigner formalism is proposed, based on which the minimization problem is designed and analyzed for the ground state, providing a new perspective for ground state calculations of the Wigner function. Employing density functional theory, a gradient flow model is built upon the energy functional to obtain the ground state Wigner function representing the entire many-body system. Numerically, a parallelizable algorithm is developed using the operator splitting method and the Fourier spectral collocation method, whose numerical complexity of a single iteration is [Formula: see text]. Numerical experiments demonstrate the anticipated accuracy, encompassing the one-dimensional system with up to [Formula: see text] particles and the three-dimensional system with defect, showcasing the potential of our approach to large-scale simulations and computations of systems with defect. To the best of our knowledge, this is the first deterministic method for calculating the ground state Wigner function of the three-dimensional many-body system.
Abstract. In this paper, a gradient flow model is presented for conducting ground state calculations in the Wigner formalism of many-body systems in the framework of density functional theory. Theoretically, an energy functional in the Wigner formalism is proposed, based on which the minimization problem is designed and analyzed for the ground state, providing a new perspective for ground state calculations of the Wigner function. Employing density functional theory, a gradient flow model is built upon the energy functional to obtain the ground state Wigner function representing the entire many-body system. Numerically, a parallelizable algorithm is developed using the operator splitting method and the Fourier spectral collocation method, whose numerical complexity of a single iteration is [Formula: see text]. Numerical experiments demonstrate the anticipated accuracy, encompassing the one-dimensional system with up to [Formula: see text] particles and the three-dimensional system with defect, showcasing the potential of our approach to large-scale simulations and computations of systems with defect. To the best of our knowledge, this is the first deterministic method for calculating the ground state Wigner function of the three-dimensional many-body system.
Abstract When considering quantum systems in phase space, the Wigner function is used as a function of quasi-probability density. Finding the Wigner function is related to the calculation of a Fourier transform of a certain composition of wave functions of the corresponding quantum system. As a rule, knowledge of the Wigner function is not the final goal, and the calculation of mean values of different quantum characteristics of the system is required. An explicit solution of the Schrödinger equation can be obtained only for a narrow class of potentials, therefore, in the majority of cases, numerical methods must be used to find wave functions. As a result, finding the Wigner function is related to the numerical integration of grid wave functions. When considering a one-dimensional system, the calculation of N2 Fourier integrals of the grid wave function is needed. To provide the necessary accuracy for wave functions, corresponding to the higher states of a quantum system, a larger number of grid nodes is needed. The goal of this work was to construct the numerical-analytical method for finding the Wigner function, which allows the number of computational operations to be considerably reduced. The quantum systems with polynomial potentials were considered, for which the Wigner function is represented as a series in some known functions.
This paper presents a numerical method for computing the ground states of the three‑dimensional Gross-Pitaevskii equation （GPE） that includes the Lee-Huang-Yang （LHY） correction.The LHY term in the Gross-Pitaevskii energy functional is of the form |ψ|5 <?fx-imagestate width="4.40266657" height="3.89466691"?> <?fx-imagestate width="4.40266657" height="3.89466691"?> and accounts for quantum fluctuations in Bose-Einstein condensates （BECs）.First， the continuous projected gradient flow is discretized with a Lagrange multiplier by using a Fourier pseudo-spectral method in space and a backward Euler scheme in time， obtaining a linearly implicit backward Euler scheme （PGF-BESP）.The key idea is to implicitly determine the Lagrange multiplier through an orthogonal condition， thereby guaranteeing mass conservation （the normalization constraint） and providing an explicit expression for the Lagrange multiplier directly from the orthogonal condition.This preserves the two fundamental properties of the continuous normalized gradient flow with a Lagrange multiplier： Mass conservation and energy dissipation.Next， we carry out a rigorous theoretical analysis of the scheme and prove that the discrete energy is monotonically non-increasing under a suitable condition on the time step size.We then perform numerical experiments to verify the theoretical results， and the results show that mass is indeed conserved and the energy decays monotonically.Finally， we use the proposed method to compute ground state configurations for different potential terms and different interaction parameters.We find that as the interaction parameter increases， the overall size of the ground state density distribution also increases， and under the optical lattice potential， multi-layer structures are observed.In summary， both theoretical and numerical results show that the proposed PGF-BESP method provides a stable， efficient and easy to implement numerical tool for ground state computations of GPE with LHY correction.
We showcase the advantages of orbital-free density-potential functional theory (DPFT), a more flexible variant of Hohenberg-Kohn density functional theory. DPFT resolves the usual trouble with the gradient-expanded kinetic energy functional by facilitating systematic semiclassical approximations in terms of an effective potential energy that incorporates all interactions. With the aid of two systematic approximation schemes we demonstrate that DPFT is not only scalable, universally applicable in both position and momentum space, and allows kinetic and interaction energy to be approximated consistently, but can also compete with highly accurate, yet restricted, methods. As two- and three-dimensional geometries are extensively covered elsewhere, our focus here is on one-dimensional settings, with semiclassical observables systematically derived from both the Wigner function formalism and a split-operator approach. The high quality of our results for Fermi gases in Morse potentials invites the use of DPFT for describing more exotic systems, such as trapped large-spin fermion mixtures with contact or dipole-dipole interactions.
The question of which non-interacting Green's function 'best' describes an interacting many-body electronic system is both of fundamental interest as well as of practical importance in describing electronic properties of materials in a realistic manner. Here, we study this question within the framework of Baym-Kadanoff theory, an approach where one locates the stationary point of a total energy functional of the one-particle Green's function in order to find the total ground-state energy as well as all one-particle properties such as the density matrix, chemical potential, or the quasiparticle energy spectrum and quasiparticle wave functions. For the case of the Klein functional, our basic finding is that minimizing the length of the gradient of the total energy functional over non-interacting Green's functions yields a set of self-consistent equations for quasiparticles that is identical to those of the quasiparticle self-consistent GW (QSGW) (van Schilfgaarde et al 2006 Phys. Rev. Lett. 96 226402-4) approach, thereby providing an a priori justification for such an approach to electronic structure calculations. In fact, this result is general, applies to any self-energy operator, and is not restricted to any particular approximation, e.g., the GW approximation for the self-energy. The approach also shows that, when working in the basis of quasiparticle states, solving the diagonal part of the self-consistent Dyson equation is of primary importance while the off-diagonals are of secondary importance, a common observation in the electronic structure literature of self-energy calculations. Finally, numerical tests and analytical arguments show that when the Dyson equation produces multiple quasiparticle solutions corresponding to a single non-interacting state, minimizing the length of the gradient translates into choosing the solution with largest quasiparticle weight.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .