  We present a comprehensive study on the magnetic structure, dynamics, and
phase evolution in the single-phase double perovskite $La_2CoMnO_6$. The mixed
valence state due to oxygen deficiency is verified by X-ray photoelectron
spectroscopy, and confirms a double ferromagnetic transition observed in DC
magnetization. Neutron diffraction reveals that the magnetic structure is
dominated by long-range ferromagnetic ordering, which is further corroborated
by a critical exponents analysis of the paramagnetic to ferromagnetic phase
transition. An analysis of the magnetization dynamics by means of linear and
nonlinear ac magnetic susceptibilities marks the presence of two distinct
cluster glass-like states that emerge at low temperatures. The isothermal
entropy change as a function of temperature and magnetic field (H) is exploited
to investigate the mechanism of stabilization of the magnetic phases across the
H-T phase diagram. In the regime of the phase diagram where thermal energy is
sufficiently low, regions of competing interactions due to local disorder
become stabilized and display glass-like dynamics. The freezing mechanism of
clusters is illustrated using a unique probe of transverse susceptibility that
isolates the effects of the local anisotropy of the spin clusters. The results
are summarized in a new H-T phase diagram of $La_2CoMnO_6$ revealed for the
first time from these data.
