A new variational wave function is proposed for the ground state of Bose liquids. It consists of a Jastrow function multiplied by exponentials of three-particle functions. The form of the wave function is deduced from a formal analysis carried out for the weakly interacting Bose gas characterized by a strength parameter $\ensuremath{\lambda}$. By using the wave function proposed here, the energy obtained is exact to $O({\ensuremath{\lambda}}^{4})$. Several useful and straightforward applications of the new method to liquid helium and Coulomb gases are suggested.
The method of correlated basis functions (CBF) is examined for the special case of a weakly interacting Bose gas. Using the Hugenholtz-Pines theory, we compute the ground-state energy exactly to fourth order in the interaction strength. The resulting expression is compared term by term with the unperturbed ground-state energy computed in the CBF using a Jastrow function as the correlating factor. We find that the three leading orders are all accounted for by the Jastrow function, while beginning in the fourth order only selected terms are included. The use of a correlated wave function in effect corresponds to summations of sellected diagrams to all orders. In particular, the ring diagrams and the ladder diagrams are most susceptible to these summations. In a separate paper we shall examine effects of perturbation in the CBF.
Continuing the diagrammatic analysis of the method of correlated basis functions (CBF) for the weakly interacting Bose gas, as reported in the preceding paper, we take off from the unperturbed problem, which is solved essentially by a variational method using a Jastrow correlating factor as the trial wave function. We compute low-order corrections to the unperturbed energy, using a set of correlated basis functions. It is found that the simplest second-order correction in the CBF accounts for all discrepancies (between the Hugenholtz-Pines theory and the unperturbed CBF calculation) which arise in the fourth order of the interaction strength $O({\ensuremath{\lambda}}^{4})$. It is also demonstrated indirectly that the Jastrow function sums rings and ladders at least up to $O({\ensuremath{\lambda}}^{5})$.
The ground-state correlation functions of a one-dimensional homogeneous Bose system described by the Lieb-Liniger Hamiltonian are investigated by using exact quantum Monte Carlo techniques. This article is an extension of a previous study published in Phys. Rev. A {\bf 68}, 031602 (2003). New results on the local three-body correlator as a function of the interaction strength are included and compared with the measured value from three-body loss experiments. We also carry out a thorough study of the short- and long-range behavior of the one-body density matrix.
Many useful properties of dilute Bose gases at ultralow temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of $N=l\ifmmode\times\else\texttimes\fi{}n$ bosons is derived by symmetrizing the
n-fold product of an
l-particle quantum state. Quantum correlations of the
l-particle state ``spread out'' to any subset of the
N bosons by symmetrization. The particle-correlated states can be simulated efficiently for large
N, because their parameter spaces, which depend on
l, do not grow with
n. Here we formulate and develop in great detail the pure-state case for
l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. 73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time
O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-
N limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-
N approximate form, and comparing with the exact ground state. For
N=1000, the relative errors of the ground-state energy for both cases are within ${10}^{\ensuremath{-}5}$ over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.