  We study the performance of permanent states (the bosonic counterpart of the
Slater determinant state) as approximating functions for bosons, with the
intention to develop variational methods based upon them. For a system of $N$
identical bosons, a permanent state is constructed by taking a set of $N$
arbitrary (not necessarily orthonormal) single-particle orbitals, forming their
product and then symmetrizing it. It is found that for the one-dimensional
Bose-Hubbard model with the periodic boundary condition and at unit filling,
the exact ground state can be very well approximated by a permanent state, in
that the permanent state has high overlap (at least 0.96 even for 12 particles
and 12 sites) with the exact ground state and can reproduce both the ground
state energy and the single-particle correlators to high precision. For a
generic model, we have devised a greedy algorithm to find the optimal set of
single-particle orbitals to minimize the variational energy or maximize the
overlap with a target state. It turns out that quite often the ground state of
a bosonic system can be well approximated by a permanent state by all the
criterions of energy, overlap, and correlation functions. And even if the error
is apparent, it can often be remedied by including more configurations, i.e.,
by allowing the variational wave function to be a combination of multiple
permanent states. The algorithm is used to study the stability of a
two-particle system, with great success. All these suggest that permanent
states are very effective as variational wave functions for bosonic systems,
and hence deserve further studies.
