Higher-Order Topological In-Bulk Corner State in Pure Diffusion Systems

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 Schrödinger equation and diffusion equation, research on higher-order topological insulators has 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.” Because of the anti-Hermitian properties of diffusive Hamiltonian, we investigate the thermal behavior of these corner states through theoretical calculation, simulation, and experiment. Furthermore, we study the different thermal behaviors 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.

Figure 1

Diffusive 2D SSH model with in-bulk corner states. (a) Schematic diagram of sphere-rod structure with four constant temperature boundaries set at the room temperature (i.e., fixed boundary condition). Thermal diffusivities of red thin and green thick rods are $D_1$ and $D_2$. Blue bars denote constant temperature heat reservoirs. The light brown box indicates one unit cell with four spheres. The upper inset is the first Brillouin zone of diffusive 2D SSH model. The lower inset is schematic diagram for the equivalent tight-binding model of one unit cell. The spheres in the black dashed boxes are stimulated as the bulk excitation discussed in Fig. 4. (b) The band structure of diffusive 2D SSH model. (c) The fixed boundary condition spectrum of diffusive 2D SSH model. The inset is the enlarged view of four in-bulk corner states and their adjacent bulk states. (d),(e) The mode distributions of two in-bulk corner states, which are indicated in the inset of Fig. 1.

Figure 2

In-bulk corner state in the nontrivial lattice. (a) Schematic diagram of the experimental setup. (b) Normalized temperature evolution of corner sphere for the nontrivial and trivial lattices. Here, $T_i$ is the excited temperature of corner sphere and $T_f$ is the temperature at the thermal equilibrium, i.e., room temperature. (c),(d) Theoretical temperature distributions for the nontrivial lattice at (c) 90 s and (d) 180 s. (e),(f) Simulated temperature distributions for the nontrivial lattice at (e) 90 s and (f) 180 s. (g),(h) Experimental temperature distributions for the nontrivial lattice at (g) 90 s and (h) 180 s.

Figure 3

Temperature distribution of trivial lattice. (a),(b) Theoretical temperature distributions for the trivial lattice at (a) 90 s and (b) 180 s. (c),(d) Simulated temperature distributions for the trivial lattice at (c) 90 s and (d) 180 s. (e),(f) Experimental temperature distributions for the trivial lattice at (e) 90 s and (f) 180 s.

Figure 4

In-bulk corner state and in-gap corner state. (a) Schematic diagram of sphere-rod structure for anisotropic diffusive 2D SSH model. Thermal diffusivities of red thin, green thick, and orange thicker rods are $D_{x1}=D_{y1}$, $D_{y2}$, and $D_{x2}$, respectively. The spheres in the black dashed boxes are stimulated as the bulk excitation. (b) The fixed boundary condition spectrum of anisotropic diffusive 2D SSH model. The inset is the enlarged view of four thermal corner states and their adjacent edge states. (c) The difference of decay rates (retrieved from the curve of normalized temperature with time) between corner spheres excitation and corner and bulk spheres excitation with different $T_{\mathrm{bulk}}$’s. Here $T_{\mathrm{bulk}}$ is the excitation temperature of bulk spheres. $R_{0,2y}=3.5\mathrm{mm}$. (d) The difference of decay rates (retrieved from the curve of normalized temperature with time) between corner spheres excitation and corner and bulk spheres excitation with different $R_{0,2y}$’s. The inset is the gap size between the second and third bands with the varying of $R_{0,2y}$. Here $T_{\mathrm{bulk}}=323.2\mathrm{K}$.