  An improved particle swarm optimization algorithm is proposed and its
superiority over standard particle swarm optimization algorithm is tested on
two typical benchmark functions. By employing this algorithm to search for the
magnetic ground states of the Hubbard model on the real-space square lattice
with finite size based on the mean-field approximation, two new magnetic
states, namely the double striped-type antiferromagnetic state and the triple
antiferromagnetic state, are found. We further perform mean-field calculations
in the thermodynamical limit to confirm that these two new magnetic states are
not a result of a finite-size effect, where the properties of the double
striped-type antiferromagnetic state are also presented.
