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arXiv:1908.02084 · Acta Physica Polonica A

Efficient Construction of Many-Body Fock States Having the Lowest Energies

Andrzej Chrostowski, Tomasz Sowiński

Published Sep 1, 2019• 18 citations

Abstract

To perform efficient many-body calculations in the framework of the exact diagonalization of the Hamiltonian one needs an appropriately tailored Fock basis built from the single-particle orbitals.The simplest way to compose the basis is to choose a finite set of single-particle wave functions and find all possible distributions of a given number of particles in these states.It is known, however, that this construction leads to very inaccurate results since it does not take into account different many-body states having the same energy on equal footing.Here we present a fast and surprisingly simple algorithm for generating the many-body Fock basis built from many-body Fock states having the lowest non-interacting energies.The algorithm is insensitive to details of the distribution of single-particle energies and it can be used for an arbitrary number of particles obeying bosonic or fermionic statistics.Moreover, it can be easily generalized to a larger number of components.Taking as a simple example the system of two ultra-cold bosons in an anharmonic trap, we show that exact calculations in the basis generated with the algorithm are substantially more accurate than calculations performed within the standard approach.

DOI: 10.12693/aphyspola.136.566

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To perform efficient many-body calculations in the framework of the exact diagonalization of the Hamiltonian one needs an appropriately tailored Fock basis built from the single-particle orbitals.The simplest way to compose the basis is to choose a finite set of single-particle wave functions and find all possible distributions of a given number of particles in these states.It is known, however, that this construction leads to very inaccurate results since it does not take into account different many-body states having the same energy on equal footing.Here we present a fast and surprisingly simple algorithm for generating the many-body Fock basis built from many-body Fock states having the lowest non-interacting energies.The algorithm is insensitive to details of the distribution of single-particle energies and it can be used for an arbitrary number of particles obeying bosonic or fermionic statistics.Moreover, it can be easily generalized to a larger number of components.Taking as a simple example the system of two ultra-cold bosons in an anharmonic trap, we show that exact calculations in the basis generated with the algorithm are substantially more accurate than calculations performed within the standard approach.

A procedure is discussed that searches for the best description of the eigenstates of a Hamiltonian of a finite quantum many-body system in terms of a selected set of physically relevant configurations. The procedure resorts to iterative sequences of diagonalizations in spaces of very reduced size. Each diagonalization provides an energy-based importance measure that governs the selection of the configurations to be included in the states. The procedure is strictly variational and preserves the symmetries of the Hamiltonian throughout the iterative process. We report on some test applications to the Na

{}_{8}

metal cluster. A series of calculations is performed on a Hartree-Fock basis for a number of orbitals ranging from 5 to 20. In the case of 5 (10) orbitals, we compare the low-lying spectra generated with our procedure to complete (truncated) configuration interaction calculations. The procedure accurately reproduces the results in the complete space. In the most complex cases of 15 and 20 orbitals (the latter involving a many-particle space of 10

{}^{11}

states), we construct the ground state and study its structure. The analysis highlights the efficiency of the method in generating the most appropriate model space for the description of individual eigenstates of the Hamiltonian.

We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.

Any exact eigenstate with a definite momentum of a many-body Hamiltonian can be written as an integral over a symmetry-broken function $\ensuremath{\Phi}$. For two particles, we exactly express $\ensuremath{\Phi}$ in terms of (single-particle) orbitals for all energy levels and any interparticle interaction. Especially for the ground state, $\ensuremath{\Phi}$ is given by the simple Hartree-Fock and Hartree Ans\"atze for fermions and bosons, respectively. Implications for several and many particles as well as a numerical example for interacting bosons are provided.

Standard analytical construction of the many-body wave function of interacting particles in one dimension, beyond mean-field theory, is based on the Jastrow approach. The many-body interacting ground state is build up from the ground state of the non-interacting system and the product of solutions of the corresponding interacting two-body problem. However, this is possible only if the center-of-mass motion is decoupled from the mutual interactions. In our work, based on the general constraints given by contact nature of the atom-atom interactions, we present an alternative approach to the standard construction of the pair-correlation wave function. Within the proposed ansatz, we study the many-body properties of trapped bosons as well as fermionic mixtures and we compare these predictions with the exact diagonalization approach in a wide range of particle numbers, interaction strengths, and different trapping potentials.