  Topological phases are greatly enriched by including non-Hermiticity. While
most works focus on the topology of the eigenvalues and eigenstates, how
topologically nontrivial non-Hermitian systems behave in dynamics has only
drawn limited attention. Here, we consider a breathing honeycomb lattice known
to emulate the quantum spin Hall effect and exhibits higher-order corner modes.
We find that non-reciprocal intracell couplings introduce gain in one
pseudo-spin subspace while loss with the same magnitude in the other. In
addition, non-reciprocal intracell couplings can also suppress the spin mixture
of the edge modes at the boundaries and delocalize the higher-order corner
mode. Our findings deepen the understanding of non-Hermitian topological phases
and bring in the spin degree of freedom in manipulating the dynamics in
non-Hermitian systems.
