# Valley Physics in High Magnetic Fields: Correlations, Broken Symmetry Phases and Transport Signatures

The interplay between quantum Hall ordering and spontaneously broken "internal" symmetries in two-dimensional electron systems with spin or pseudospin degrees of freedom gives rise to a variety of interesting phenomena, including novel phases, phase transitions, and topological excitations, often with striking experimental signatures. I will focus on a class of multi-valley systems, where the symmetries include point-group operations that rotate spatial axes and simultaneously permute the indices of "valleys", i.e. the minima in the electronic dispersion. Generically, these discrete symmetries can be broken at finite temperature in clean systems, leading in the simplest examples to "valley Ising ferromagnets" with accompanying spatial nematic order. Weak disorder destroys this order through domain formation; however, the resulting state still asymptotically exhibits the QHE. I will discuss transport properties in the ordered and disordered regimes, and their relevance to recent experiments in AlAs quantum wells at Landau level filling $\nu=1$. I will also discuss a the case of Si quantum wells, where the discrete point-group symmetry is intertwined with a set of continuous valley rotations and leads to a rich phase structure for $1\leq \nu\leq 6$ that can be understood as arising from a valley analog of the mechanism of `order by disorder'. References: D.A. Abanin, S.A. Parameswaran, S.A Kivelson, S.L. Sondhi, Phys. Rev. B 82, 035428 (2010). A. Kumar, S.A. Parameswaran and S.L. Sondhi, Phys. Rev. B 88, 045133 (2013). A. Kumar, S.A. Parameswaran and S.L. Sondhi, work in progress.

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