Microscopic theory of a quantum Hall Ising nematic: Domain walls and disorder

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 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 trilayer graphene with trigonal warping.