  The bilayer Hubbard model with electron-hole doping is an ideal platform to
study excitonic orders due to suppressed recombination via spatial separation
of electrons and holes. However, suffering from the sign problem, previous
quantum Monte Carlo studies could not arrive at an unequivocal conclusion
regarding the presence of phases with clear signatures of excitonic
condensation in bilayer Hubbard models. Here, we develop a determinant quantum
Monte Carlo (DQMC) algorithm for the bilayer Hubbard model that is
sign-problem-free for equal and opposite doping in the two layers, and study
excitonic order and charge and spin density modulations as a function of
chemical potential difference between the two layers, on-site Coulomb
repulsion, and inter-layer interaction. In the intermediate coupling regime and
in proximity to the SU(4)-symmetric point, we find a biexcitonic condensate
phase at finite electron-hole doping, as well as a competing $(\pi,\pi)$ charge
density wave (CDW) state. We extract the Berezinskii-Kosterlitz-Thouless (BKT)
transition temperature from superfluid density and a finite size scaling
analysis of the correlation functions, and explain our results in terms of an
effective biexcitonic hardcore boson model.
