  The calculation of the ground state and thermodynamics of mass-imbalanced
Fermi systems is a challenging many-body problem. Even in one spatial
dimension, analytic solutions are limited to special configurations and
numerical progress with standard Monte Carlo approaches is hindered by the sign
problem. The focus of the present work is on the further development of methods
to study imbalanced systems in a fully non-perturbative fashion. We report our
calculations of the ground-state energy of mass-imbalanced fermions using two
different approaches which are also very popular in the context of the theory
of the strong interaction (Quantum Chromodynamics, QCD): (a) the hybrid Monte
Carlo algorithm with imaginary mass imbalance, followed by an analytic
continuation to the real axis; and (b) the Complex Langevin algorithm. We cover
a range of on-site interaction strengths that includes strongly attractive as
well as strongly repulsive cases which we verify with non-perturbative
renormalization group methods and perturbation theory. Our findings indicate
that, for strong repulsive couplings, the energy starts to flatten out,
implying interesting consequences for short-range and high-frequency
correlation functions. Overall, our results clearly indicate that the Complex
Langevin approach is very versatile and works very well for imbalanced Fermi
gases with both attractive and repulsive interactions.
