Quantum Hall Skyrmion Lattices at $g \approx 0$

We investigate the classical ground state configurations of a collection of skyrmions in the limit of vanishing L\'ande $g$ factor. We show that in this limit, the skyrmions which are large and overlapping, behave qualitatively differently from small skyrmions. We investigate the system in various regimes of spin stiffness. We find that in all regimes, at $g=0$, the ground state configuration is a rectangular skyrmion lattice with N\'eel orientational ordering. The charge distribution has a higher symmetry of a face-centered rectangular lattice and the $z$-component of the spin polarization is zero. We also present general argument for the vanishing of spin polarization at $g=0$ at finite temperature and in the presence of disorder and discuss some experimental consequences of our results.