Order by Disorder and by Doping in Quantum Hall Valley Ferromagnets

Abstract

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 $\hat{I}½$ in the range $\hat{I}½=0-6$. This six-valley system has a large $[SU(2){]}^3⋊D_3$ 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 $T\to 0$ the $[SU(2){]}^3$ continuous symmetry also breaks: completely near $\hat{I}½=3$, to a residual $[U(1){]}^2\times SU(2)$ near $\hat{I}½=2$ and $4$ and to a residual $U(1)\times [SU(2){]}^2$ near $\hat{I}½=1$ and $5$. Interestingly, the symmetry breaking near $\hat{I}½=2,4$ and $\hat{I}½=3$ 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.

5+3 pages, 3 figures