  We explore nonequilibrium quantum heat transport in nonlinear bosonic systems
in the presence of a non-Kerr-type interaction governed by hyper-parametric
oscillation due to two-photon hopping between the two cavities. We estimate the
thermodynamic response analytically by constructing the $su(2)$ algebra of the
nonlinear Hamiltonian and predict that the system exhibits a negative
excitation mode. Consequently, this specific form of interaction enables the
cooling of the system by inducing a ground state transition when the number of
particles increases, even though the interaction strength is small. We
demonstrate a transition of the heat current numerically in the presence of
symmetric coupling between the system and bath and show long relaxation times
in the cooling phase. We compare with the Kerr-type Bose-Hubbard form of
interaction induced via cross-phase modulation, which does not exhibit any such
transition. We further compute the nonequilibrium heat current in the presence
of two baths at different temperatures and observe that the cooling effect for
the non-Kerr-type interaction persists. Our findings may help in the
manipulation of quantum states using the system's interactions to induce
cooling.
