  We investigate heat transport through a one-dimensional open coupled scalar
field theory, depicted as a network of harmonic oscillators connected to
thermal baths at the boundaries. The non-Hermitian dynamical matrix of the
network undergoes a stability-to-instability transition at the exceptional
points as the coupling strength between the scalar fields increases. The open
network in the unstable regime, marked by the emergence of inverted oscillator
modes, does not acquire a steady state, and the heat conduction is then
unbounded for general bath couplings. In this work, we engineer a unique bath
coupling where a single bath is connected to two fields at each edge with the
same strength. This configuration leads to a finite steady-state heat
conduction in the network, even in the unstable regime. We also study general
bath couplings, e.g., connecting two fields to two separate baths at each
boundary, which shows an exciting signature of approaching the unstable regime
for massive fields. We derive analytical expressions for high-temperature
classical heat current through the network for different bath couplings at the
edges and compare them. Furthermore, we determine the temperature dependence of
low-temperature quantum heat current in different cases. Our study will help to
probe topological phases and phase transitions in various quadratic Hermitian
bosonic models whose dynamical matrices resemble non-Hermitian Hamiltonians,
hosting exciting topological phases.
