With the rapid growth of the computational resources, quantum Monte Carlo (QMC) methodology has become a powerful tool for numerical simulations, especially for lattice effective field theory. Those QMC simulations have successfully described the physics of the few- and many-body systems. In this thesis, we investigate the Fermi systems with balanced and unbalanced populations of up and down spins using QMC with ab initio techniques. We also present a newly developed method called eigenvector continuation (EC) and its promising applications to some numerically unavoidable problems, like the sign problem in Monte Carlo simulations. For the eigenvector continuation method, we demonstrate that although Hamiltonian is usually represented as a matrix in a linear space with enormous dimensions, the eigenvector trajectory generated by a smoothly changed Hamitonian matrix is well approximated by a low-dimensional space. We use analytic continuation theory to prove this statement and propose an algorithm to implement our method. In the simulation with strong numerical sign oscillations, we first “learn” the subspace where the trajectory is approximately spanned by a finite number of accurately computable eigenvectors and then apply eigenvetor continuation to solve the physics system where there is a severe sign problem. Our results converge rapidly as we include more eigenvectors. The results show that for the same computational cost, the EC method reduces errors by an order of magnitude compared to the direct calculations in cases when we have a strong sign problem. In L × L × L cubic lattices with various box sizes, we study the ground-state properties of fermionic many-body system in the unitary limit. The universal parameter (Bertsch parameter ξ) is calculated with high accuracy and the result is extrapolated to infinite volume. We characterize the superfluid phase in this system by calculating the off-diagonal long-range order of the two-body density matrix, and the condensate fraction α, which is calculated to be 0.43(1). In addition, we study the properties of the superfluid pairs. Thepair size ζp is found to be proportional to k−1 and the ratio ζp/k−1 is 1.93(9).
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