Convergence properties of Fock-space based approaches in strongly correlated Fermi gases : a dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
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Date
2019
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Massey University
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Abstract
The main objective of this thesis is the effcient numerical description of strongly
correlated quantum gases. Due to the complex many-body structure of the wave
function, usually, numerical methods are required for its computation. The exact
diagonalization approach is considered, where the energies and the wave functions
are obtained by diagonalizing the Hamiltonian in a many-body basis. The dimension
of the space increases combinatorially with the number of particles and the
number of single-particle basis functions, which limits the characterization of fewbody
systems to intermediate interactions. One of the main components of the
convergence rate originates from the particle-particle interaction itself. The bare
contact interaction introduces a singularity in the wave function at the particleparticle
coalescence point. This is responsible for the slow convergence in the
nite basis expansion in one dimension and it even causes pathological behavior
in higher dimensions.
Firstly, the Gaussian interaction potential is examined as an alternative pseudopotential.
After the description of the accurate calculation of the s-wave scattering
length of this potential, the convergence properties are investigated. As
this function is smooth, by construction the wave function is free from any singularity
implying an exponentially fast convergence rate. If the resolution of the
basis set is not fine enough, the finite-range pseudopotential is indistinguishable
from the pathological contact potential. Through the example of few particles in
a two-dimensional harmonic trap, we show that in order to reach the necessary
resolution, the number of harmonic-oscillator single-particle basis functions must
increase quadratically with the inverse characteristic length of the pseudopotential.
This scaling property combined with the combinatorial growth of the many-body
space makes the physically realistic short-range potentials computationally inaccessible.
We have also applied the so-called transcorrelated approach, where the singular
part of the wave function is isolated in a Jastrow-type factor. This factor can be
transformed into the Hamiltonian reducing the irregularity of the eigenfunction and improving the convergence rate. We will show through the example of the
homogeneous gas in one dimension that this transformation efficiently improves
the convergence from M⁻¹ to M⁻³, where M is the number of the single-particle
plane-wave basis functions.
Description
Listed in 2019 Dean's List of Exceptional Theses
Keywords
Many-body problem, Numerical solutions, Wave functions, Configuration space, Electron gas, Dean's List of Exceptional Theses