  Recently, the attractive Kane-Mele-Habbard (KMH) model on a honeycomb lattice
at half filling has been studied in two papers: PRB 99, 184514 (2019) and PRB
94, 104508 (2016). The authors of the first one presented the phase diagram
which interpolates the trivial and non-trivial topological states. However, the
next-nearest-neighbor (NNN) hopping term has been neglected, although it is
several orders of magnitude stronger than the internal spin-orbit coupling. We
use the mean-field approximation to derive the phase diagram of the attractive
KMH model with NNN hoping at half filling. The phase diagram without and the
phase diagram with NNN hopping are significantly different in the non-trivial
topological region. The possibility to have superconducting instability in the
attractive KMH model has been analyzed in the second paper within the T-matrix
approximation. The question that naturally arises here is about the
contributions due to the bubble diagrams, which are included in the
Bethe-Salpeter (BS) equation, but neglected by the T-matrix approximation. To
answer this question, we apply the BS formalism to calculate the slope of the
Goldstone mode and the corresponding sound velocity. We found 4% difference
between the values of the sound velocity provided by the T-matrix approximation
and the BS equation. This small difference confirm previously reported result
that close to the phase transition boundary the bubble-diagram contributions
are not important.
