Field Theory of Skyrme Lattices in Quantum Hall Ferromagnets

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 %SMG superconductor-insulator and the non-zero temperature phase transitions.