Heat generation in a quantum dot side-coupled to a Majorana bound state pair

We employ the small polaron transformation and the nonequilibrium Green's function method to derive an analytical expression for the heat generation in a quantum dot side-coupled to a topological superconducting loop hosting a Majorana bound state pair. Our main findings are threefold: (i) Gate-voltage control: Under symmetric bias, the heat generation profile evolves from a double-peak to a single-peak structure as the side-coupled strength increases; under extreme asymmetric bias, a flux phase of ϕ = π induces a pronounced enhancement of heat generation at specific dot levels. (ii) Bias-voltage control: The differential electrothermal conductance ∂Qd/∂V exhibits clear and well- resolved peaks, and in terms of phase control, it offers complementary information of the Majorana bound state pair signatures compared to the differential conductance ∂I/∂V . (iii) Magnetic-flux control: The heat-generation signals display a robust crossover from 2π to 4π periodicity, tunable by the Majorana overlap energy and temperature, serving as a distinct phase fingerprint to distinguish the Majorana bound state pair from trivial ring structures. Overall, the heat generation and its derivative serve as a useful complement to conventional charge transport measurements for detecting Majorana bound states, offering an additional probe across all three tuning dimensions.