  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.
