  Various unusual behaviors of artificial materials are governed by their
topological properties, among which the edge state at the boundary of a
photonic or phononic lattice has been captivated as a popular notion. However,
this remarkable bulk-boundary correspondence and the related phenomena are
missing in thermal materials. One reason is that heat diffusion is described in
a non-Hermitian framework because of its dissipative nature. The other is that
the relevant temperature field is mostly composed of modes that extend over
wide ranges, making it difficult to be rendered within the tight-binding theory
as commonly employed in wave physics. Here, we overcome the above challenges
and perform systematic studies on heat diffusion in thermal lattices. Based on
a continuum model, we introduce a state vector to link the Zak phase with the
existence of the edge state, and thereby analytically prove the thermal
bulk-boundary correspondence. We experimentally demonstrate the predicted edge
states with a topologically protected and localized heat dissipation capacity.
Our finding sets up a solid foundation to explore the topology in novel heat
transfer manipulations.
