Lattice field theory is a useful tool for studying strongly interacting theories in condensed matter physics.A prominent example is the unitary Fermi gas: a two-component system of fermions interacting with divergent scattering length.With Monte Carlo methods this system can be studied from first principles.In the presence of an imbalance (unequal number of particles in the two components) a sign problem arises, which makes conventional algorithms inapplicable.We will show how to apply reweighting techniques to generalise the recently developed worm algorithm to the imbalanced case, and present results for the critical temperature, the energy per particle, the chemical potential and the contact density for equal, as well as unequal number of fermions in the two spin components.
This dissertation discusses the experimental study of strong fermionic interactions in an optical lattice. Ultracold samples of
40K are prepared in an intricate experimental platform using the combination of various optical and evaporative cooling techniques. The atoms are adiabatically loaded into an optical lattice to realize isolated pair states. Various tools for the coherent manipulation of spin and orbital degrees of freedom of the atoms in the lattice are developed. The inter-particle interactions between pairs are tuned using magnetic Feshbach resonances. Spectroscopic measurements of resonantly enhanced and controllable s-wave and p-wave interaction energies are reported. The measurements extend to the unitary limit of interactions characterized by diverging scattering lengths and universal observables. These measurements show excellent agreements to an exact solution for two harmonically confined atoms interacting via a pseudopotential and numerical solutions using an \textit{ab initio} interaction potential. For p-wave interactions, exquisite experimental control is demonstrated by measuring Rabi oscillations between non-interacting and strongly interacting atomic pairs, which are used to accurately measure highly sensitive properties such as the two-body wavefunction overlap. With the full suppression of three-body loss processes, the lifetime of p-wave interacting pairs is investigated. The experiments demonstrate that losses of p-wave interacting pairs are limited by the intrinsic lifetime of the free-space molecular dimer, resulting in the observation of lifetimes that are fifty times larger than previous measurements in
40K. The observation, control, and comprehensive understanding of strong fermionic interactions demonstrated in this thesis illuminate a path towards the assembly of new many-body states of matter.
We formulate a path-integral Monte Carlo algorithm for simulating lattice systems consisting of fictitious particles governed by a generalized exchange statistics. This method, initially proposed for continuum systems, introduces a continuous parameter
ξ in the partition function that interpolates between bosonic (
ξ=1) and fermionic (
ξ=−1) statistics. We generalize this approach to discrete lattice models and apply it to the two-dimensional Hubbard model of fictitious particles, including the Bose- and Fermi-Hubbard models as special cases. By combining reweighting and
ξ-extrapolation techniques, we access both half-filled and doped regimes. In particular, we demonstrate that the method remains effective even in strongly correlated, doped systems where the fermion sign problem hinders conventional quantum Monte Carlo approaches. Our results validate the applicability of the fictitious particle framework on lattice models and establish it as a promising tool for sign-problem mitigation in strongly interacting fermionic systems.