Selectable diffusion direction with topologically protected edge modes

Topological insulators, known for their robust edge modes, offer significant potential for controlling wave and diffusion phenomena. Previous research has explored thermal localization and decay in one-dimensional (1D) systems based on the Su-Schrieffer-Heeger (SSH) model. Higher-dimensional systems, such as those utilizing the quantum spin Hall effect (QSHE), have shown promise in wave control, including localization and one-way propagation. Recent work has also applied these concepts to diffusion systems, demonstrating thermal localization and decay rate control in 1D and 2D structures using models like SSH and Kagome lattices. However, these studies primarily focused on localization and decay rate, lacking control over heat transport and thermal polarization. While Hermitian systems incorporating thermal distribution and fluid convection have achieved unidirectional heat transport, they often involve complex flow paths unsuitable for broad scalability. This research aims to address these limitations by investigating the QSHE in a 2D thermal diffusion system, aiming to achieve selectable directional heat transport and thermal polarization with a simpler design.

The application of topological properties to various wave systems, including electromagnetic, acoustic, and mechanical systems, has garnered substantial interest. The SSH model is a common approach for generating edge modes, enabling wave control in 1D systems. The QSHE, with its higher degrees of freedom, offers greater control in higher dimensions, facilitating localization and one-way wave propagation. Studies have explored applying these concepts to diffusion systems, demonstrating thermal localization and robust decay in 1D structures based on the SSH model. Higher-order topological corner modes have expanded these capabilities to 2D systems, showing the localization of high- or low-temperature spots. Other 2D models, such as Kagome lattices, have also been investigated, revealing topological locking decay rates. However, these primarily focused on localization and decay rates, not on heat transport or polarization. Recent proposals have used Hermitian systems combining thermal distribution and fluid convection (skin effect) to achieve unidirectional heat transport, but these often lack scalability. The current study focuses on the largely unexplored potential of QSHE-based topological states with high degrees of freedom to control diffusion phenomena with a simple diffusivity design.

The researchers designed a structure composed of periodically aligned honeycomb-shaped unit cells. Each unit cell contains six sites connected by beams with effective diffusivities D₁ and D₂. The thermal diffusion within a unit cell is represented by a 6x6 effective diffusivity matrix. The topological and ordinary states of the unit cells were controlled by adjusting the ratio r = D₁/D₂. Diagonalization of the effective diffusivity matrix yielded the eigenvalue spectrum, revealing the existence of a topological surface state when all diffusivities are equal (r=1). For r>1 and r<1, a bandgap and doubly degenerate modes emerged. Analysis of eigenfunctions revealed the temperature distributions of these modes, identifying dipole and quadrupole modes. The inversion of the order of these modes indicated a topological phase transition. To demonstrate topologically protected edge modes, a supercell with a boundary between topological and ordinary states was created. The spectrum of the supercell showed topological edge modes with different polarizations, localized at the boundary. Analysis of heat transfer between sites in the unit cells revealed unique directional heat balance for different edge modes. Time evolution of temperature distributions for these modes was simulated using COMSOL Multiphysics. The eigenfunctions were converted to temperature distributions, and the system was excited by setting these temperatures at specific sites. The simulations revealed thermal polarization along the x-axis for one mode and the y-axis for another, confirming the selectable diffusion direction. The influence of antiphase modes, obtained by 180° rotation of the temperature distribution, was also investigated, demonstrating the potential for further control of diffusion direction. Finally, the robustness of the edge modes to defects was tested by removing beams from some sites; results indicated that the edge states remain relatively unaffected by these disorders. The COMSOL simulations used aluminum as the material, with specific thermal conductivity values adjusted to obtain the desired effective diffusivities. Adiabatic boundary conditions were applied to eliminate external thermal influences. Simulations tracked temperature changes over time.

The study found that topologically protected edge modes exist in a thermal diffusion system with a honeycomb structure. By adjusting the ratio of effective diffusivities (r = D₁/D₂), the system transitions between topological and ordinary states. The eigenfunctions of the system revealed distinct temperature distributions corresponding to dipole and quadrupole modes. The inversion of these mode orders signifies the topological phase transition. A supercell model demonstrated topologically protected edge modes localized at the boundary between topological and ordinary states. These edge modes exhibit a unique directional heat balance, allowing for the selection of macroscopic thermal diffusion direction. COMSOL Multiphysics simulations confirmed that these edge modes induce thermal polarization along the x-axis for one mode and the y-axis for another, validating the ability to select the diffusion direction. The inclusion of antiphase modes further expanded the control over the diffusion direction. The system proved robust against defects, maintaining its directional thermal diffusion capabilities even when beams were removed from the structure. Analysis of the temperature decay rates showed a significantly higher uniformity and robustness for the edge modes compared to bulk modes, highlighting the topological protection against defects. The temperature decay rates are slower for the edge modes indicating a longer-lived topological protection.

The findings demonstrate the successful control of thermal diffusion direction using topologically protected edge modes in a 2D honeycomb structure. This surpasses previous research, which primarily focused on controlling the decay rate of thermal diffusion. The robustness of these edge modes to defects showcases their practical applicability. The ability to select the diffusion direction using different edge modes, and further refine it with antiphase modes, opens up new possibilities for thermal management and temperature control. The observed uniformity in decay rates between edge modes, even in the presence of defects, highlights the advantage of topological protection in maintaining consistent thermal diffusion. The results suggest broader applicability to other diffusion phenomena, beyond thermal diffusion. The similarity in the governing equations between wave and diffusion systems indicates the potential for further exploration of these topological effects in various applications.

This research successfully demonstrates selectable directional thermal diffusion using topologically protected edge modes in a honeycomb lattice. The system's robustness to defects and ability to control diffusion direction through mode selection offers significant potential for applications in thermal management and beyond. Future work could explore the extension of this approach to other diffusion phenomena, such as ionic transport, and a deeper understanding of the relationship between wave and diffusion systems in the context of topological effects.

The study focused on thermal diffusion, and the direct applicability to other diffusion phenomena requires further investigation. The time scale over which topological protection is maintained is limited by the diffusion process itself. While the system shows robustness to defects, the extent of this robustness with more significant or different types of defects could be further explored. The model used in the simulations is a simplified representation of a real-world system; real-world implementations might encounter additional complexities.