  We study disorder-induced magnetism within the Gutzwiller approximation
applied to the t-J model relevant for cuprate superconductors. In particular,
we show how disorder generates magnetic phases by inducing local droplets of
antiferromagnetic order which eventually merge, and form a quasi-long range
ordered state in the underdoped regime. We identify two distinct
disorder-induced magnetic phases of this type depending on the strength of the
scatterers. For weak potential scatterers used to model dopant disorder, charge
reorganization may push local regions in-between the impurities across the
magnetic phase boundary, whereas for strong scatterers used to model
substitutional ions, a local static magnetic moment is formed around each
impurity. We calculate the density of states and find a remarkably universal
low-energy behavior largely independent of both disorder and magnetization.
However, the magnetic regions are characterized by larger (reduced)
superconducting gap (coherence peaks) and a sub-gap kink in the density of
states.
