  Motivated by recent experimental processes, we systemically investigate
strongly correlated spin-1 ultracold bosons trapped in a three-dimensional
optical lattice in the presence of an external magnetic field. Based on a
recently developed bosonic dynamical mean-field theory (BDMFT), we map out
complete phase diagrams of the system for both antiferromagnetic and
ferromagnetic interactions, where various phases are found as a result of the
interplay of spin-dependent interaction and quadratic Zeeman energy. For
antiferromagnetic interactions, the system demonstrates competing magnetic
orders, including nematic, spin-singlet and ferromagnetic insulating phase,
depending on longitudinal magnetization, whereas, for ferromagnetic case, a
ferromagnetic-to-nematic-insulating phase transition is observed for small
quadratic Zeeman energy, and the insulating phase demonstrates the nematic
order for large Zeeman energy. Interestingly, at low magnetic field and finite
temperature, we find an abnormal multi-step condensation of the strongly
correlated superfluid, i.e. the critical condensing temperature of the $m_F=-1$
component with antiferromagnetic interactions demonstrates an increase with
longitudinal magnetization, while, for ferromagnetic case, the Zeeman component
$m_F = 0$ demonstrates a local minimum for the critical condensing temperature,
in contrast to weakly interacting cases.
