Order by Disorder and by Doping in Quantum Hall Valley Ferromagnets

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We examine the Si(111) multi-valley 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)]³ D₃ 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". We also exhibit modestly simpler analogs in the four-valley Si(110) system.