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We report the application of the nonlinear $\sigma$ model to study the
multi-skyrmion problem in the quantum Hall ferromagnet system. We make use of
a first-principle calculation to derive an analytical form for the
inter-skyrmionic interaction to show that the ground state of the system can
be described by a ferromagnet triangular Skyrme lattice near $\nu=1$ where
skyrmions are extremely dilute and a continuous transition into
antiferromagnet square lattice occurs by increasing the skyrmion density and
therefore $[\nu-1]$. Using these results we demonstrate that the transition
for a triangular to a square lattice which was previously derived, using the
Hartree-Fock method, can also be seen in the field theory picture. We
investigate the possibility that the skyrmions bound in pair to make a
bi-skyrmion triangular lattice when the Zeeman energy is extremely small. We
show that the energy of a skyrmion with charge $Q$ is less than the energy of
$Q$ skyrmions each with charge one when the short range interaction among them
is considered. By taking the quantum fluctuations into account, we also argue
the possibility of the existence of a superconductor-insulator and the
non-zero temperature phase transitions.
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