Effective field theories at the edges of topological insulators

We consider interacting two-dimensional time reversal invariant topological insulators with time reversal preserving (non-magnetic) disorder near their edges. These systems are known to fail to exhibit quantized spin currents even when the fermion-fermion interactions are weak, as was recently shown within Hartree-Fock theory (Novelli et al., Phys. Rev. Lett. 122, 016601). Here, we investigate the microscopic origin of the lack of quantization by considering the most relevant few-body processes at the edges of the system, and find that fermion-impurity resonances are responsible for backscattering. We build two different effective field theories for the interacting edge modes in the system near the Fermi energy that eliminate the bulk in two different ways: 1) by keeping the inhomogeneity explicitly and 2) by accounting for the width of the resonance in the dispersion relations.