  Compared with conventional topological insulator that carries topological
state at its boundaries, the higher-order topological insulator exhibits
lower-dimensional gapless boundary states at its corners and hinges. Leveraging
the form similarity between Schrodinger equation and diffusion equation,
researches on higher-order topological insulators have been extended from
condensed matter physics to thermal diffusion. Unfortunately, all the corner
states of thermal higher-order topological insulator reside within the band
gap. Another kind of corner state, which is embedded in the bulk states, has
not been realized in pure diffusion systems so far. Here, we construct
higher-dimensional Su-Schrieffer-Heeger models based on sphere-rod structure to
elucidate these corner states, which we term ``in-bulk corner states". Due to
the anti-Hermitian properties of diffusive Hamiltonian, we investigate the
thermal behaviour of these corner states through theoretical calculation,
simulation, and experiment. Furthermore, we study the different thermal
behaviours of in-bulk corner state and in-gap corner state. Our results would
open a different gate for diffusive topological states and provide a distinct
application for efficient heat dissipation.
