Heat Diffusion Invariant.

Topological invariants have been utilized profoundly to classify the geometric features of electronic, photonic, and phononic structures. However, no invariant has ever been conceptualized to classify the functional properties of diverse thermal structures. Here, we formulate a heat diffusion invariant to reveal the explicit correlation between thermal functionality and diffusivity. The value goes beyond establishing a unified framework for various thermal metamaterials. The heat diffusion invariant offers a forward paradigm for precisely achieving freeform transient thermal metamaterials with isotropic thermal conductivity. We verify the competitiveness and robustness experimentally via a freeform transient thermal cloak. The heat diffusion invariant also paves the way for other functions, such as transient thermal illusion, and applies to thermal convection and radiation. These findings could open up an unprecedented invariant-based gateway to thermal management, metamaterial design, and nonequilibrium energy and mass transport.