  The checkerboard lattice is a two-dimensional non-trivial structure usually
seen as a planar version of the pyrochlore lattice. This geometry supports a
two-band insulating electronic system with Chern topology induced by a complex
hoppping parameter. Inspired by the recent advances in the topology of
non-Hermitian systems, in this work we study a non-Hermitian version of the
topological checkerboard lattice. The complex band structure and Berry
curvature are calculated. In the insulating phase, the Chern number is the same
as in the Hermitian version, but the Hall conductivity is no longer quantized.
The dependence of the Hall conductivity with the non-Hermitian parameters is
investigated. The non-Hermiticity can be seen as a result of dissipation caused
by coupling the system to the environment, so this study casts light on the
topology of open systems in condensed matter physics.
