  We investigate the attractive Fermi polaron problem in two dimensions using
non-perturbative Monte Carlo simulations. We introduce a new Monte Carlo
algorithm called the impurity lattice Monte Carlo method. This algorithm
samples the path integral in a computationally efficient manner and has only
small sign oscillations for systems with a single impurity. As a benchmark of
the method, we calculate the universal polaron energy in three dimensions in
the scale-invariant unitarity limit and find agreement with published results.
We then present the first fully non-perturbative calculations of the polaron
energy in two dimensions and density correlations between the impurity and
majority particles in the limit of zero range interactions. We find evidence
for a smooth crossover transition from fermionic quasiparticle to molecular
state as a function of interaction strength.
