  Topological phases in spinless non-Hermitian models have been widely studied
both theoretically and experimentally in some artificial materials using
photonics, phononics and magnon. In this work, we investigate the interplay
between non-Hermitian loss and gain and non-Abelian gauge potential realized in
a two-component superconducting circuit. In our model, the non-Hermiticity
along only gives rise to trivial gain and loss to the states; while the
non-Abelian gauge along gives rise to flying butterfly spectra and associated
edge modes, which in photonics can be directly measured by the intensity of
photons at the boundaries. These two terms do not commute, and their interplay
can give rise to several intriguing non-Hermitian phases, including the fully
gapped quantum spin Hall (QSH) phase, gapless QSH phase, trivial gapped phase
and gapless metallic phase. The bulk-edge correspondence is absent and we find
that during the closing of energy gap in the gapped QSH phase, the system
enters the gapless QSH phase regime which still supports two
counter-propagating edge modes. We have also unveiled the intriguing role of
non-Hermiticity on the chiral symmetry and time-reversal symmetry of the
Hermitian models, which can be applied to other physical models.
