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arXiv:cond-mat/0510436 · Physical Review A

Simple renormalization-group approximation of the ground-state properties of interacting bosonic systems

Roman Werpachowski, Jerzy Kijowski

Published Jun 21, 2007• 0 citations
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Abstract

We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.

DOI
10.1103/physreva.75.062113

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We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.
The variational determination of the two-boson reduced density matrix is described for a one-dimensional system of N N harmonically trapped bosons interacting via contact interaction, where N N ranges from 2 2 to 10^4 10 4 . The ground-state energies are calculated, and compared to existing methods, including the analytic case (for N=2) N = 2 ) and mean-field approaches such as the one-dimensional Gross-Pitaevskii equation and its variations. Structural properties such as the density and correlation functions are also derived, including the behaviour of the correlation function when boson coordinates coincide. This collectively demonstrates the capacity of the reduced density matrix method to accurately calculate collective ground-state properties of bosonic systems comprising few to many bosons, including the cross-over region between these extremes, across a large range of interaction strengths.
We formulate a quantum Monte Carlo (QMC) method for calculating the ground state of many-boson systems. The method is based on a field-theoretical approach, and is closely related to existing fermion auxiliary-field QMC methods which are applied in several fields of physics. The ground-state projection is implemented as a branching random walk in the space of permanents consisting of identical single-particle orbitals. Any single-particle basis can be used, and the method is in principle exact. We apply this method to an atomic Bose gas, where the atoms interact via an attractive or repulsive contact two-body potential parametrized by the s-wave scattering length. We choose as the single-particle basis a real-space grid. We compare with exact results in small systems, and arbitrarily-sized systems of untrapped bosons with attractive interactions in one dimension, where analytical solutions exist. Our method provides a way to systematically improve upon the mean-field Gross-Pitaevskii (GP) method while using the same framework, capturing interaction and correlation effects with a stochastic, coherent ensemble of non-interacting solutions. to study the role of many-body correlations in the ground state, we examine the properties of the gas, such as the energetics, condensate fraction, and the density and momentum distributions as a function of the number of particles and the scattering length, both in the homogenous and trapped gases. Results are presented for systems with up to 1000 bosons. Comparing our results to the mean-field GP results, we find significant departure from mean field at large positive scattering lengths. The many-body correlations tend to increase the kinetic energy and reduce the interaction energy compared to GP. In the trapped gases, this results in a qualitatively different behavior as a function of the scattering length. Possible experimental observation is discussed.
We consider systems of a small number of interacting bosons confined to harmonic potentials in one and two dimensions. By exact numerical diagonalization of the many-body Hamiltonian we determine the low-lying excitation energies and the ground-state energy and density profile. We discuss the dependence of these quantities on both interaction strength ggg and particle number NNN. The ground-state properties are compared to the predictions of the Gross-Pitaevskii equation, and the agreement is surprisingly good even for relatively low particle numbers. We also calculate the specific heat based on the obtained energy spectra.
The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method known as reduced Hamiltonian interpolation (RHI) [D. A. Mazziotti and D. R. Herschbach, Phys. Rev. Lett. 83, 5185 (1999)]. In the RHI, the N-particle Hamiltonian is represented through a sequence of p-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. Assimilating ideas from N representability and dimensional interpolation, the RHI technique interpolates over the number p of quasiparticles to determine the N-particle energy with close upper and lower bounds. With the Hellmann-Feynman theorem, we extend the RHI to compute the two-particle reduced density matrix (2RDM) as well as the energy. We examine both the computational advantages of RHI in comparison with traditional methods and the possibility of extending the RHI to treat fermion correlation. Applied to bosons with harmonic interactions as well as a two-level system, the RHI technique yields more than 99% of the correlation energy and an accurate correlation correction for the elements of the 2RDM.
We formulate a quantum Monte Carlo (QMC) method for calculating the ground state of many-boson systems. The method is based on a field-theoretical approach, and is closely related to existing fermion auxiliary-field QMC methods which are applied in several fields of physics. The ground-state projection is implemented as a branching random walk in the space of permanents consisting of identical single-particle orbitals. Any single-particle basis can be used, and the method is in principle exact. We illustrate this method with a trapped atomic boson gas, where the atoms interact via an attractive or repulsive contact two-body potential. We choose as the single-particle basis a real-space grid. We compare with exact results in small systems and arbitrarily sized systems of untrapped bosons with attractive interactions in one dimension, where analytical solutions exist. We also compare with the corresponding Gross-Pitaevskii (GP) mean-field calculations for trapped atoms, and discuss the close formal relation between our method and the GP approach. Our method provides a way to systematically improve upon GP while using the same framework, capturing interaction and correlation effects with a stochastic, coherent ensemble of noninteracting solutions. We discuss various algorithmic issues, including importance sampling and the back-propagation technique for computing observables, and illustrate them with numerical studies. We show results for systems with up to N approximately 400 bosons.