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Higher-order topological phases in non-Hermitian photonics revolutionize the understanding of wave propagation and modulation, which lead to hierarchical states in open systems. However, intrinsic insulating properties endorsed by the lattice symmetry of photonic crystals fundamentally confine the robust transport only at explicit system boundaries, letting alone the flexible reconfiguration in hierarchical states at arbitrary positions. Here, we report a dynamic topological platform for creating the reconfigurable hierarchical bound states in heat transport systems and observe the robust and nonlocalized higher-order states in both the real-and imaginary-valued bands. Our experiments showcase that the hierarchical features of zero-dimension corner and nontrivial edge modes occur at tailored positions within the system bulk states instead of the explicit system boundaries. Our findings uncover the mechanism of non-localized hierarchical non-trivial topological states and offer distinct paradigms for diffusive transport field management. [ABSTRACT FROM AUTHOR]

The topological physics has sparked intensive investigations into topological lattices in photonic, acoustic, and mechanical systems, powering counterintuitive effects otherwise inaccessible with usual settings. Following the success of these endeavors in classical wave dynamics, there has been a growing interest in establishing their topological counterparts in diffusion. Here, we propose an additional real-space dimension in diffusion, and the system eigenvalues are transformed from "imaginary" to "real." By judiciously tailoring the effective Hamiltonian with coupling networks, localized and delocalized topological modes are realized in heat transfer. Simulations and experiments in active thermal lattices validate the effectiveness of the proposed theoretical strategy. This approach can be applied to establish various topological lattices in diffusion systems, offering insights into engineering topologically protected edge states in dynamic diffusive scenarios. [ABSTRACT FROM AUTHOR]