Biasing topological charge injection in topological matter

We explore the interplay between topologies in momentum and real spaces to formulate a thermodynamic description of nonequilibrium injection topological charges under external bias. show that edge modes engendered by momentum-space topology can play functional role connecting reservoirs bulk transport space. illustrate our general results with two examples: spin-torque skyrmions an electrically-biased integer quantum Hall system, vortex $p\,+\,i\,p$ superconductor coupled heat reservoirs. Based on universal fractional entropy Majorana zero bound vortices, their controllable proposed this work could provide route for creating manipulating fermions.