Magnetization Instability in a Two-Dimensional System

The spin magnetization of a system of fermions confined to two dimensions, and subject to a strong magnetic field, has been predicted to exhibit a first-order phase transition as a function of its Zeeman energy. We present strong experimental evidence that such a paramagnetic to ferromagnetic phase transition occurs at Landau level filling $\nu =4\mathrm{}$ in the two-dimensional hole gas. We demonstrate that our data are not explained by perturbative effects such as Landau level anticrossing, and show that exchange coefficients measured at odd filling factors are consistent with the predicted size of the gap at $\nu =4\mathrm{}$.