  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.
