Full configuration interaction quantum Monte Carlo and its application to the lattice Bose polaron at quantum criticality : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics, Centre for Theoretical Chemistry and Physics, Institute for Advanced Study, Massey University, Auckland, New Zealand

Abstract

This thesis provides an investigation of the application of the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm to ultracold atom problems. This includes working on the algorithm, understanding the sign problem, and applying it to the study of strongly-correlated lattice polaron physics.

The thesis conceptually consists of two parts. In the first, we discuss the FCIQMC algorithm, which is a modern variant of projector Monte Carlo and can be used for calculating the ground states or low-lying excited states of quantum Hamiltonians. We present our implementation of the algorithm in the Julia programming language and discuss the details of the algorithm and data structures employed.

Next, we turn to a treatment of the FCIQMC sign problem, which arises when the number of walkers employed by FCIQMC is insufficient to capture the sign structure of the ground state eigenvector of the Hamiltonian. We present a new scalar model that accurately reconstructs the dynamic behaviour of FCIQMC and shows us that as the number of walkers increases, the walker population switches from a sign-incoherent regime to a sign-coherent one in way that is reminiscent of a continuous phase transition.

In the second part, we focus on the application of FCIQMC to the Bose-Hubbard model (BHM) with an impurity near quantum criticality. We use FCIQMC to guide and validate a diagrammatic theory based on the quantum Gutzwiller method, which describes the polaron by treating interactions between the impurity and the elementary excitations in the BHM. This theory shows that at strong interactions, the polaron reflects the state of the host system and as such, polarons could be used as quantum sensors in experiments.

Lastly, we show numerical evidence that the polaron energy --- the energy shift caused by introducing an impurity to the BHM --- is scale invariant near the phase transition and that it exhibits critical phenomena not yet explained by theory. Furthermore, we show that the polaron energy is linked to the impurity-boson density-density correlation function and that it exhibits a flattening near the phase transition. This flattening indicates that in the thermodynamic limit, the length scales associated with the polaron diverge.