  We study the the interplay between spontaneously broken valley symmetry and
spatial disorder in multivalley semiconductors in the quantum Hall regime. In
cases where valleys have anisotropic electron dispersion a previous
long-wavelength analysis [Phys. Rev. B 82, 035428 (2010)] identified two new
phases exhibiting the QHE. The first is the quantum Hall Ising nematic (QHIN),
a phase with long-range orientational order manifested in macroscopic transport
anisotropies. The second is the quantum Hall random-field paramagnet (QHRFPM)
that emerges when the Ising ordering is disrupted by quenched disorder,
characterized by a domain structure with a distinctive response to a valley
symmetry-breaking strain field. Here we provide a more detailed microscopic
analysis of the QHIN, which allows us to (i) estimate its Ising ordering
temperature; (ii) study its domain-wall excitations, which play a central role
in determining its properties; and (iii) analyze its response to quenched
disorder from impurity scattering, which gives an estimate for domain size in
the descendant QHRFPM. Our results are directly applicable to AlAs
heterostructures, although their qualitative aspects inform other ferromagnetic
QH systems, such as Si(111) heterostructures and bilayer graphene with trigonal
warping.
