Roton instability of the spin-wave excitation in the fully polarized quantum Hall state and the phase diagram at v = 2

We consider the effect of interaction on electrons confined to two dimensions at Landau level filling v = 2, with the specific aim of determining the range of parameters where the fully polarized state is stable. We calculate the charge-and the spin-density collective modes in the random-phase approximation (RPA) including vertex corrections (also known as the time-dependent Hartree-Fock approximation), and treating the Landau level mixing accurately within the subspace of a single particle-hole pair. It is found that the spin-wave excitation mode of the fully polarized state has a roton minimum which deepens as a result of the interaction-induced Landau level mixing, and the energy of the roton vanishes at a critical Zeeman energy, signalling an instability of the fully polarized state at still lower Zeeman energies. The feasibility of the experimental observation of the roton minimum in the spin-wave mode and its softening will be discussed. The spin-and charge-density collective modes of the unpolarized state are also considered, and a phase diagram for the v = 2 state as a function of r_s and the Zeeman energy is obtained.