Impossibility of refrigeration and engine operation in minimal qubit repeated-interaction models

We investigate the operation of a qubit as a quantum thermal device within the repeated interaction framework, allowing for strong system-bath coupling and finite interaction times. We analyze two minimal models: an alternating-coupling setup, in which the qubit sequentially interacts with hot and cold baths, and a simultaneous-coupling setup, where both baths interact with the qubit during each collision. For the alternating model, we obtain an exact analytical solution for the limit-cycle state, valid for arbitrary coupling strengths and collision durations. Using this solution, we rigorously prove a no-go theorem for quantum refrigeration. We further demonstrate that although work can be generated locally at individual system-bath contacts, the total work over a cycle is always nonpositive, precluding engine operation. In the absence of work, the model describes pure heat conduction, for which we derive a closed-form expression for the heat current and show that it exhibits a nonmonotonic turnover behavior. The simultaneous-coupling model is analyzed perturbatively. In the short-collision-time limit, it reproduces the same steady-state behavior as the alternating model, reinforcing the generality of the constraints identified. Our results establish fundamental limitations on qubit-based quantum thermal machines operating under Markovian repeated interactions and highlight the need for enriched models to realize functional quantum thermal devices.