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Order by Disorder and by Doping in Quantum Hall Valley Ferromagnets

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 $[SU(2)]^3\rtimes D_3$ symmetry in the limit where the magnetic length is much larger than the lattice constant. We find that the discrete ${D}_3$ factor breaks over a broad range of fillings at a finite temperature transition to a discrete nematic phase. As $T \rightarrow 0$ the $[SU(2)]^3$ continuous symmetry also breaks: completely near $ν=3$, to a residual $[U(1)]^2\times SU(2)$ near $ν=2$ and $4$ and to a residual $U(1)\times [SU(2)]^2$ 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.