Figure 1

(a) A ferromagnetic layer deposited on the surface of a topological insulator; (b) the blue (light) disk indicates the region in momentum space bounded by a Fermi surface with Fermi energy ${\mu }_F$. Small arrows on the Fermi surface indicate an attractive interaction between electrons (red balls) with momenta ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$ and the angle between them is ${\varphi }_{{\mathbf{p}}_1,{\mathbf{p}}_2}={\varphi }_{{\mathbf{p}}_2}-{\varphi }_{{\mathbf{p}}_1}$. (c) Feynman diagrammatic representation of Bethe-Salpeter equation and the interaction in direct and exchange channels, and in (d) we show the regions with the non-Fermi liquid behavior and pairing instability at zero temperature. The horizontal axis stands for the frequency of transverse magnon excitations with propagator $D_T(q)$. In the low frequency regime, the excitations are highly damped and $\mathrm{\Sigma }(\omega )\sim {\omega }^{2/3}$, while in the opposite regime up to an energy scale given by ${\alpha }_2$ the magnons are described by a bare propagator $D_T^0(q)$ and $\mathrm{\Sigma }(\omega )\sim {\omega }^{1/2}$ dominates over the linear term. In both regimes the system is unstable to Amperean pairing. For some experimentally relevant parameters the propagator of magnons falls into the damped regime shown by a blue solid disk with strong instability to Amperean pairing.