  One of the most important applications of quantum mechanics is the
thermodynamic description of quantum gases. Despite the fundamental importance
of this topic, a comprehensive description of the thermodynamic properties of
non-Hermitian quantum gases is still lacking. Here, we investigate the
properties of bosonic and fermionic non-Hermitian systems at finite
temperatures. We show that non-Hermitian systems exihibit oscillations both in
temperature and imaginary time. As such, they can be a possible platform to
realize an imaginary time crystal (iTC) phase. The Hatano-Nelson model is
identified as a simple lattice model to reveal this effect. In addition, we
show that the conditions for the iTC to be manifest are the same as the
conditions for the presence of disorder points, where the correlation functions
show oscillating behavior. This analysis makes clear that our realization of
iTC is effectively a way to filter one specific Matsubara mode. In this
realization, the Matsubara frequency, that enters as a mathematical tool to
compute correlation functions for finite temperatures, can be measured
experimentally.
