Geometric heat pumping under continuous modulation in thermal diffusion

Berry (geometric) phase has attracted a lot of interest and permeated into all aspects of physics including photonics, crystal dynamics, electromagnetism and heat transfer since it was discovered, leading to various unprecedented effects both in classical and quantum systems, such as Hannay angle, quantum Hall effect, orbital magnetism and Thouless pumping. Heat pumping is one of the most prominent and fantastic application of geometric phase in heat transport. Here we derive a general heat pumping theory based on classical diffusion equation and continuous modulation of system parameters in macroscopic thermal diffusion system and obtain a formula which is reminiscent of contact between Berry phase and the Berry curvature. Furthermore, we discuss two cases of non-trivial zero heat flux after one cycle which is fundamentally different from the trivial zero heat flux generated by static zero heat bias in physical nature. Then we analyze the dependence of the effect on the system thermal parameters, including some counterintuitive phenomenon. Finally, under the guidance of this theory, we conduct an experiment to demonstrate the accuracy and effectiveness of our theory and observe the heat pumping effect regardless of the presence and the absence of the thermal bias between two ports of system. In general, our work clearly derives the universal form of heat pumping theory under arbitrary form of the modulation in the macroscopic thermal diffusion system, this is of great significance for better heat energy transport, heat manipulation and so on. It also establishes the foundation of achieving other non-reciprocity devices or topological devices with the aid of spatiotemporal modulation.