Hierarchical topological states in thermal diffusive networks

The integration of topological concepts into electronic energy band theory has been a transformative development in condensed matter physics. Since then, this paradigm has broadened its reach, extending to a variety of physical systems, including open ones. In this study, we employ analogs of the generalized $n$-dimensional Su-Schrieffer-Heeger model, a cornerstone in understanding topological insulators and higher-order topological states, to unveil a dimensional hierarchy of topological states within thermal diffusive networks. Unlike their electronic counterparts, the topological states in these networks are characterized by confined temperature profiles of dimension $(n\ensuremath{-}d)$ with constant diffusive rates, where $n$ represents the system's dimension and $d$ is the order of the topological state. Our findings demonstrate the existence of topological corner states in thermal diffusive systems up to $n=3$, along with surface and hinge states. We also identify and discuss an intermediate-order topological phase in the case $n=3$, characterized by the presence of hinge states but the absence of corner states. Furthermore, our work delves into the influence of chiral symmetry in these thermal networks, particularly focusing on topological thermal states with a near-zero diffusion rate. This research lays the foundation for advanced thermal management strategies that utilize topological states in multiple dimensions.