Quarter-quantized thermal Hall effect with parity anomaly

We show that in the proximity of s-wave superconductors, the magnetic\ntopological surface states can transform into Majorana surface state, featuring\na single gapless Majorana cone with parity anomaly when the superconducting\npairing gap matches the surface magnetization gap. The emergence of $N=1/2$\nMajorana chiral edge current is observed at the boundaries between the gap\nregion and the gapless region. Additionally, in systems with a single gapless\nMajorana cone, a quarter-quantized thermal Hall conductance appears under the\ndephasing. By mapping the system to a conductor-network model, we identify the\nappearance of 1/4 chiral heat channels as the cause of the quarter-quantized\nHall thermal conductance. We observe the stability of this quarter-quantized\nthermal conductance under temperature variations, serving as a distinctive\nfeature indicating the presence of a single gapless Majorana cone in the\nsystem. Our models can be experimentally realized using magnetic topological\ninsulators or iron-based superconductors.\n