  We study excitations of the local field (locsitons) in nanoscale
two-dimensional (2D) lattices of strongly interacting resonant atoms and
various unusual effects associated with them. Locsitons in low-dimensional
systems and the resulting spatial strata and more complex patterns on a scale
of just a few atoms were predicted by us earlier [A. E. Kaplan and S. N.
Volkov, Phys. Rev. Lett., v. 101, 133902 (2008)]. These effects present a
radical departure from the classical Lorentz-Lorenz theory of the local field
(LF), which assumes that the LF is virtually uniform on this scale. We
demonstrate that the strata and patterns in the 2D lattices may be described as
an interference of plane-wave locsitons, build an analytic model for such
unbounded locsitons, and derive and analyze dispersion relations for the
locsitons in an equilateral triangular lattice. We draw useful analogies
between one-dimensional and 2D locsitons, but also show that the 2D case
enables locsitons with the most diverse and unusual properties. Using the
nearest-neighbor approximation, we find the locsiton frequency band for
different mutual orientations of the lattice and the incident field. We
demonstrate a formation of distinct vector locsiton patterns consisting of
multiple vortices in the LF distribution and suggest a way to design finite 2D
lattices that exhibit such patterns at certain frequencies. We illustrate the
role of lattice defects in supporting localized locsitons and also demonstrate
the existence of "magic shapes", for which the LF suppression at the exact
atomic resonance is cancelled.
