  We study repulsive Hubbard and t-J type systems on a square lattice (long
believed to capture certain quintessential aspects of the high temperature
superconductors). These models (alongside the parent compounds of the high
temperature superconductors) are antiferromagnetic in the absence of hole
doping. As we illustrate, a unifying underlying principle for the dynamics of
holes introduced by doping rationalizes the emergence of nonuniform electronic
structures- "stripes" and possible pairing tendencies therein. Specifically,
our analysis invokes the following (numerically verified) sublattice parity
principle: a strong antiferromagnetic background forces injected holes to hop
in steps of two such that they always remain on the same sub lattice. When
applied to a domain wall in an antiferromagnet, this simple principle naturally
gives rise to (bond centered) stripes. We demonstrate that the holes are
self-consistently localized on stripes. Extending this picture, we then show
that the holes on a stripe favor the formation of pairs on neighboring rungs or
sites. Throughout this work much emphasis is placed on the problem of a two leg
ladder immersed in a staggered magnetic field. Although we will focus on the
square lattice, our considerations may be extended to similar electronic
structures appearing in other models on bipartite lattices when these exhibit
antiferromagnetic correlations with an underlying sublattice structure.
