Order by disorder and by doping in quantum Hall valley ferromagnets

We examine the Si(111) multivalley quantum Hall system and show that it exhibits an exceptionally rich interplay of broken symmetries and quantum Hall ordering already near integer fillings ν in the range ν=0-6. This six-valley system has a large [SU(2)]3D3 symmetry in the limit where the magnetic length is much larger than the lattice constant. We find that the discrete D3 factor breaks over a broad range of fillings at a finite-temperature transition to a discrete nematic phase. As T→0, the [SU(2)]3 continuous symmetry also breaks: completely near ν=3, to a residual [U(1)]2×SU(2) near ν=2 and 4, and to a residual U(1)×[SU(2)]2 near ν=1 and 5. Interestingly, the symmetry breaking near ν=2,4 and ν=3 involves a combination of selection by thermal fluctuations known as "order by disorder" and a selection by the energetics of Skyrme lattices induced by moving away from the commensurate fillings, a mechanism we term "order by doping.