Thermodynamic precision in nonequilibrium critical quantum systems

Inspired by recent advances in critical quantum heat engines, we investigate heat transport near the critical point of a Lipkin-Meshkov-Glick model coupled to two bosonic reservoirs. In the steady-state and thermodynamic limit, we analyze the thermodynamic precision of the model by examining the scaling of the relative fluctuation and the correspondence between the thermodynamic uncertainty relation ratio and the purity of the reduced density matrix of the central system. Furthermore, we observe a pronounced minimum in the thermodynamic uncertainty relation ratio at the critical point. These results are also verified using Redfield equation, polaron-transformed master equation, and analytical results from the quantum jump theory. Our findings contribute to the understanding of critical thermodynamic precision and provide insights into quantum heat control in nonequilibrium phase transition systems.