  We present a theoretical study of the ground state of the BCS-BEC crossover
in dilute two-dimensional Fermi gases. While the mean-field theory provides a
simple and analytical equation of state, the pressure is equal to that of a
noninteracting Fermi gas in the entire BCS-BEC crossover, which is not
consistent with the features of a weakly interacting Bose condensate in the BEC
limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of
the 2D mean-field theory indicates that the quantum fluctuations are much more
pronounced than those in 3D. In this work, we show that the inclusion of the
Gaussian quantum fluctuations naturally recovers the above features in both the
BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on
the boson chemical potential is recovered by the quantum fluctuations. Near the
quantum phase transition from the vacuum to the BEC phase, we compare our
equation of state with the known grand canonical equation of state of 2D Bose
gases and determine the ratio of the composite boson scattering length $a_{\rm
B}$ to the fermion scattering length $a_{\rm 2D}$. We find $a_{\rm B}\simeq
0.56 a_{\rm 2D}$, in good agreement with the exact four-body calculation. We
compare our equation of state in the BCS-BEC crossover with recent results from
the quantum Monte Carlo simulations and the experimental measurements and find
good agreements.
