The Fermi hypernetted-chain (FHNC) theory is applied to spin-density-wave states combined with a Jastrow-type correlation operator. The commonly used approximation scheme FHNC/n is found to be unsuitable for dealing with this class of correlated wave functions. A new scheme, named FHNC-n, is proposed, and numerically tested for the one-dimensional (1D) Hubbard model and the Gutzwiller correlation operator. The results obtained are in very good agreement with the available variational Monte Carlo calculation. This scheme can be used to study 2D and 3D Hubbard models and can easily be extended to treat more general correlation operators, which include triplet and quartet correlations.
Abstract The main task of microscopic many-body theory is to provide an understanding and an explanation of properties of macroscopic systems from no other information than the properties of the underlying Hamiltonian, the particle statistics, and the macroscopic geometry of the system. Variational wave functions have been a very successful technique for examining the ground state and dynamic properties of dense quantum fluids. Specifically the “optimized (Fermi)-hypernetted chain” ((F)HNC-EL) hierarchy of approximations reproduces, for simple interactions like electrons and quantum fluids, the most basic features of a self-bound many-body system, namely binding, saturation, and spinodal decomposition. The relationships between Green’s functions based perturbation theory and variational wave functions have been clarified in much detail in the past. In the language of Feynman diagrams, a self-consistent summation of ring- and ladder-diagrams is the minimum requirement for a satisfactory microscopic description of these basic features. Realistic nucleon–nucleon interactions pose further problems: The most difficult aspect of the nuclear many-body problem is caused by the form of the microscopic nucleon–nucleon interaction which depends at least on the spin, isospin, the relative orientation and angular momentum of the interacting particles. Simple variational methods as used for electrons and quantum fluids are inadequate. Correlation functions that have the same structure as the interactions lead to so-called “commutator diagrams” that have, so far, been mostly neglected. We have shown in the past that these corrections can be very important if the two-body interactions are very different in different spin- or isospin channels. To deal with the problem of realistic nuclear interactions we have combined techniques of the Jastrow–Feenberg variational method and the local parquet-diagram theory. In the language of diagrammatic perturbation theory, “commutator diagrams” can be identified with non-parquet diagrams. We examine the physical processes described by these terms and include the relevant diagrams in a way that is suggested by the Jastrow–Feenberg approach. We show that the corrections from non-parquet contributions are, at short distances, larger than all other many-body effects. We examine here neutron matter as a prototype of systems with state-dependent interactions. Calculations are carried out for neutrons interacting via the so-called
v8 v 8 version of four popular interactions. We determine the structure and effective interactions and apply the method to the calculation of the energetics, structure and dynamic properties such as the single-particle self-energy and the dynamic response functions as well as BCS pairing in both singlet and triplet states. We find that many-body correlations lead to a strong reduction of the spin-orbit interaction, and, therefore, to a suppression of the
3P2 3 P 2 and
3P2 3 P 2 -
3F2 3 F 2 gaps. We also find pairing in
3P0 3 P 0 states; the strength of the pairing gap depends sensitively on the potential model employed.
A method for the accurate evaluation of the diagonal and off-diagonal matrix elements of the Hamiltonian operator within the correlated basis function scheme is presented. Considering a basis consisting of Slater determinants, correlated with a Jastrow correlation factor, the expectation value of the kinetic energy operator has been expressed in the Jackson-Feenberg form. A diagrammatic analysis is presented for the construction and evaluation of the cluster expansions of the required quantities incorporating the Fermi hypernetted chain theory.
The ground-state wave function of an infinite system of fermions is approximated by the state-independent Jastrow ansatz. In order to optimize the pair correlation function, the Euler-Lagrange equations of the variational problem for the energy expectation value are derived. The short- and long-range behaviors of the optimum pair correlation function are discussed. Application of graphical techniques and use of rigorous results on the connection between the slope of the static structure function as $k\ensuremath{\rightarrow}0$+ and long-range Jastrow correlations allow one to prove that the optimum pair correlation function behaves like $1+O({r}^{\ensuremath{-}2})$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$. A connection is derived between the weight of the long-range correlations and the Landau parameters. As simple examples the limit of Bose statistics and the electron-gas problem are considered. The consequences of these investigations on numerical calculations and their relation to alternative expansion methods are investigated.
Correlated-basis-function (CBF) theory, which has provided a firm foundation for ab initio calculations of the properties of quantum fluids such as liquid
4He, is adapted and applied at the Jastrow-Feenberg variational level to give an optimized description of the structure and of the elementary excitations of the Ising-spin model in a transverse magnetic field. A set of trial wave functions of Hartree-Jastrow form is assumed to describe the spatial correlations present in the ordered as well as the disordered ground states, and the Feynman Ansatz is consistently adopted to represent the elementary magnon states. The CBF analysis of the spin system employs the hypernetted-chain (HNC) formalism for a substitutional binary mixture of bosons and derives HNC equations for the spatial distribution functions which determine the ground-state energy expectation value. Functional variation of this quantity with respect to the magnetic order parameter and the trial states leads to two Euler-Lagrange equations, which may be interpreted as a renormalized Hartree equation for the optimal magnetization and as a paired-magnon equation for the magnetic correlation function that is analogous to the familiar paired-phonon equation for conventional quantum fluids.Numerical calculations are based on simple cubic lattices and an optimized nearest-neighbor Ansatz for the generating pseudopotential. Results are reported on the order parameter, the energy per lattice site, the transverse magnetization, and the magnon excitation energies as functions of the coupling parameter 0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}\ensuremath{\infty} measuring the strength of the transverse magnetic field. We also present numerical results on the magnetic correlation function, the static structure function, and the correlation length. The system exhibits a second-order phase transition at a critical value ${\ensuremath{\lambda}}_{\mathit{c}}$ of the coupling strength. Our numerical calculations of the optimal order parameter yields ${\ensuremath{\lambda}}_{\mathit{c}}$\ensuremath{\simeq}5.17 for a simple cubic lattice and ${\ensuremath{\lambda}}_{\mathit{c}}$\ensuremath{\simeq}3.12 for a square planar lattice. The calculated data are in very good agreement with results derived from perturbation expansions in conjunction with Pad\'e techniques. \textcopyright{} 1996 The American Physical Society.