Twisted two-dimensional bilayer materials exhibit many exotic physical phenomena. Manipulating the twist angle between the two layers enables fine control of the physical structure, resulting in development of many novel physics, such as the magic-angle flat-band superconductivity, the formation of moire exciton and interlayer magnetism. Here, combined with analogous principles, we study theoretically the near-field radiative heat transfer (NFRHT) between two twisted hyperbolic systems. This two twisted hyperbolic systems are mirror images of each other. Each twisted hyperbolic system is composed of two graphene gratings, where there is an angle {phi} between this two graphene gratings. By analyzing the photonic transmission coefficient as well as the plasmon dispersion relation of twisted hyperbolic system, we prove that the topological transitions of the surface state at a special angle (from open (hyperbolic) to closed (elliptical) contours) can modulate efficiently the radiative heat transfer. Meanwhile the role of the thickness of dielectric spacer and vacuum gap on the manipulating the topological transitions of the surface state and the NFRHT are also discussed. We predict the hysteresis effect of topological transitions at a larger vacuum gap, and demonstrate that as thickness of dielectric spacer increase, the transition from the enhancement effect of heat transfer caused by the twisted hyperbolic system to a suppression. This technology could novel mechanism and control method for NFRHT, and may open a promising pathway for highly efficient thermal management, energy harvesting, and subwavelength thermal imaging.
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