  The interplay between non-Hermiticity and topology opens an exciting avenue
for engineering novel topological matter with unprecedented properties. While
previous studies have mainly focused on one-dimensional systems or Chern
insulators, here we investigate topological phase transitions to/from quantum
spin Hall (QSH) insulators driven by non-Hermiticity. We show that a trivial to
QSH insulator phase transition can be induced by solely varying non-Hermitian
terms, and there exists exceptional edge arcs in QSH phases. We establish two
topological invariants for characterizing the non-Hermitian phase transitions:
i) with time-reversal symmetry, the biorthogonal $\mathbb{Z}_2$ invariant based
on non-Hermitian Wilson loops, and ii) without time-reversal symmetry, a
biorthogonal spin Chern number through biorthogonal decompositions of the Bloch
bundle of the occupied bands. These topological invariants can be applied to a
wide class of non-Hermitian topological phases beyond Chern classes, and
provides a powerful tool for exploring novel non-Hermitian topological matter
and their device applications.
