The FCIQMC sign problem in the real-space Hubbard model

Abstract

Full configuration interaction quantum Monte Carlo (FCIQMC) is an electronic structure method that has been applied to a variety of ab initio molecular and solid-state systems as well as the Hubbard model in delocalised bases. In this thesis, the behaviour of FCIQMC in the Hubbard model in the real-space formulation is investigated. A special emphasis is put on the consequences of the fermionic sign problem. Firstly, a classification of Hubbard lattice geometries based on their strength of the sign problem is performed. It is discovered that the commonly used ground state of the so-called stoquastic version of the Hamiltonian is not a good predictor for the difficulty to resolve the sign problem in FCIQMC in general. The notion of size-extensive and non-size-extensive behaviour of the sign problem is established. It is shown that although the vast majority of non-trivial fermionic systems suffer from the fermion sign problem when attempting to solve them using quantum Monte Carlo (QMC) methods, there are certain system configurations in the Hubbard model systems that are sign-problem-free. In principle, this allows for the unbiased treatment of systems with very large Hilbert space sizes in FCIQMC. However, attempting to solve these systems uncovers a new systematic bias in the FCIQMC algorithm, the population control bias. This is a bias that has been observed previously in other QMC methods, like diffusion Monte Carlo. A method that allows for the removal of this bias entirely with negligible computational overhead, mainly through introducing importance sampling to FCIQMC, is presented. This allows for the calculation of ground-state energies of the one-dimensional Hubbard model with up to 150 sites at and close to half-filling in the difficult intermediate interaction regime. Also, the fundamental many-particle gaps between the ground states of the half-filled and the system with one hole are calculated for up to 102 sites. Moving to sign-problematic systems, it is shown that the usual method of controlling the sign problem in FCIQMC, the initiator method, performs poorly in weakly sign-problematic Hubbard systems. Instead, it is demonstrated how applying the newly developed importance-sampled FCIQMC together with the exact non-initiator algorithm greatly reduces the minimum number of walkers necessary to obtain an unbiased ground-state energy in real-space Hubbard models. This allows for the calculation of numerically exact ground-state energies for width-two Hubbard ladders - which exhibit a size-extensive yet very weak sign problem - in the intermediate interaction regime at half-filling and with one hole. Again, this makes the calculation of the fundamental many-particle gaps possible. Finally, to deal with full two-dimensional Hubbard systems, a way to define fixed initiator subspaces in FCIQMC based on analytic wavefunction ansatzes is presented. This leads to far superior results compared to the usual population-based initiator criterion. Additionally, the newly developed two-shift method allows for the perturbative inclusion of the entire non-initiator space. This new scheme is shown to be compatible with importance sampling. Furthermore, an extrapolation scheme to the exact ground-state energy is presented. This allows for the estimation of the ground-state energy for systems up to 32 sites in the honeycomb lattice geometry.