Selectable diffusion direction with topologically protected edge modes

The study investigates whether topologically protected edge modes, inspired by the quantum spin Hall effect (QSHE), can control not just localization and decay in diffusion systems but also the macroscopic direction of thermal diffusion. Prior work in wave systems (electromagnetic, acoustic, mechanical) has established robust edge states via SSH and QSHE models. Early applications to diffusion showed thermal localization and topologically locked decay, including higher-order (corner) modes and 2D lattices (e.g., Kagome), but lacked directional transport and polarization. Recent approaches using convection-enabled (skin-effect) Hermitian systems achieved unidirectional edge transport but at the cost of complex flow paths and limited scalability. The authors aim to realize directional, defect-robust heat polarization in a purely diffusive 2D system using a honeycomb lattice whose topology is tuned by the ratio of effective diffusivities within and between unit cells, and to demonstrate selection of diffusion direction via different edge modes.

• Topological control in classical wave systems is well established, with SSH-based edge modes enabling robust 1D wave guidance; QSHE enables higher-dimensional control, including localization and one-way propagation.
• Diffusion analogs: Experiments and theory have observed topological edge states in thermal diffusion with robust decay (SSH-like), and higher-order (corner) modes enabling localization in 2D. Kagome lattices further demonstrated locked decay rates and thermal spot localization.
• Limitations: Prior diffusion systems primarily controlled localization and decay rate, not directional transport or polarization. Convection-based (skin-effect) approaches realize edge transport but require complex flow networks that hinder scalability and generalization to higher dimensions.
• Gap: QSHE-like topological states with higher degrees of freedom have been underexplored in purely diffusive systems. This work addresses that gap by leveraging QSHE-inspired edge modes in a honeycomb lattice to select diffusion direction.

• Geometry and model: A periodic honeycomb-shaped unit cell with six disc sites connected by beams to nearest neighbors within a cell (effective diffusivity D1) and to adjacent cells (effective diffusivity D2). Beam length L = 15 mm, width w = 2 mm, disc diameter d = 20 mm. The effective diffusivities are normalized as D = λ/(ρ c L^2), where λ is thermal conductivity, ρ density, and c heat capacity.
• Topological tuning: The ratio r = D1/D2 controls the phase. For r = 1, a Dirac cone appears at the K point, indicating a topological transition point. For r > 1 and r < 1, a band gap opens with two doubly degenerate bands. Mode inversion (dipole vs quadrupole ordering) identifies trivial (ordinary, r > 1) and nontrivial (topological, r < 1) phases.
• Band/eigenfunction analysis: A 6×6 effective diffusivity matrix (anti-Hermitian Hamiltonian) is constructed in k-space for the unit cell temperatures T1–T6. Diagonalization yields eigenvalues (imaginary, representing decay rates) and eigenfunctions (site temperature profiles). For r > 1, low-polarization dipole modes (px, py) lie below the gap and quadrupole (dxy, dy2−x2) above; for r < 1 this order inverts, signaling the topological phase.
• Supercell edge-state computation: A supercell with a topological–ordinary boundary (10 units on each side) yields edge-localized modes. The band diagram shows two gap-crossing edge modes (A and B) near k∥ = 0. Mode temperatures are localized at the interface.
• Heat transfer and unit-cell heat balance: Using eigenfunction amplitudes of modes A and B, the authors compute pairwise nearest-neighbor heat transfers within and between unit cells (visualized as arrows) and sum components along x and y to obtain macroscopic heat balance per cell. Mode A yields net heat balance along x (parallel to boundary), none along y; mode B yields net along y (perpendicular to boundary), none along x.
• Time-domain simulations: Finite structure of 13×12 unit cells simulated in COMSOL (MEMS module) with aluminum properties c = 900 J kg−1 K−1, ρ = 2700 kg m−3, thickness 10 mm. Beams’ λ adjusted to realize target D1 and D2: Topological state D1 = 0.6, D2 = 0.85; Ordinary state D1 = 0.765, D2 = 0.518. Site λ = 238 W m−1 K−1. Adiabatic (no convection/radiation) boundaries. Time window 0–100 s, 2 s step. Initial temperature T0 = 293.15 K; edge mode excitation set as Tv = T0 + α Tmode with α = 100, applied to two rows near the boundary (my = 1, 2) where edge localization is strongest (amplitudes at my = 3 are ≤ 30% of my = 1).
• Antiphase modes and linear combinations: Due to structural symmetry, an antiphase mode A′ is constructed by 180° rotation of A about the y-axis; it exists as an edge mode with slightly shifted k. Mode B is axisymmetric and identical to B′. Incoherent superpositions enable tunable polarization: TAB = rγ TA + (1 − rγ) TB and TA′B = rγ TA′ + (1 − rγ) TB with 0 ≤ rγ ≤ 1.
• Defects and robustness tests: Structural defects introduced by removing beams at sites 2 and 4 in two topological unit cells (T, 6,1) and (T, 8,1). Temperatures monitored at regions adjacent to the interface (Xleft, Xright, Yabove, Ybelow) and at specific sites v = [mx, my, n]. Decay rates γs extracted from logarithmic temperature decay log[(Tv(t) − T0)/(Tv(0) − T0)].
• Practical design note: Alternative tuning of D1 and D2 via beam length adjustments (e.g., intrabeam L = 12.776 mm and interbeam L = 10.734 mm for topological; intrabeam L = 11.315 mm and interbeam L = 13.751 mm for ordinary) without changing aluminum’s intrinsic λ.

• Existence of QSHE-inspired thermal edge modes: A supercell with topological–ordinary boundary exhibits two gap-crossing edge modes (A and B) localized at the interface.
• Direction-selectable macroscopic diffusion: Mode A yields net unit-cell heat balance along x (parallel to boundary) and none along y; Mode B yields net along y (perpendicular to boundary) and none along x. Time-domain simulations confirm thermal polarization consistent with these directions (hot/cold at Xleft/Xright for A; Yabove/Ybelow for B).
• Antiphase control: The antiphase mode A′ (180° rotation of A) inverts A’s polarization (hot/cold swapped between Xleft and Xright), whereas B′ is identical to B. Hence diffusion direction can be selected among +x, −x, and ±y via A, A′, and B.
• Tunable polarization via incoherent superposition: Linear combinations TAB and TA′B with mixing ratio rγ produce linear temperature responses in Xleft, Xright, Yabove, Ybelow at t = 40 s, enabling symmetric distributions at rγ = 0.5 and continuous control of polarization strength.
• Robustness to defects: Removing beams at sites 2 and 4 in two unit cells causes only small variations in Xleft/Xright temperatures over time compared to the pristine case, indicating defect immunity due to topological protection.
• Uniform topologically protected decay: For edge mode A, site-specific decay rates at v = [5,1,4] and v = [9,1,4] are γ = 0.0858 and 0.0891 (difference 3.8%), demonstrating uniform decay across sites. For a bulk mode, decay rates are 0.0995 and 0.1182 (difference 18.8%), 4.9× larger disparity than in the edge mode.
• Temporal window: Edge-mode behavior with uniform, protected decay persists up to about 20 s for the chosen parameters, after which temperatures approach T0 and the edge mode collapses.
• Strong localization: Initial edge-mode fields are strongly localized to two rows near the boundary; amplitudes at my = 3 are ≤ 30% of those at my = 1.

The findings demonstrate that topologically protected edge modes in a honeycomb diffusion lattice can select the macroscopic direction of thermal diffusion—an ability previously lacking in diffusion-focused topological systems. Directional heat balance at the unit-cell scale translates to robust thermal polarization at the device scale, surviving structural defects. The anti-Hermitian effective Hamiltonian of the diffusion system yields imaginary eigenvalues and topologically protected decay rates, clarifying how nonstationary temperature fields manifest edge modes. Practical considerations include a trade-off between edge-mode duration and system response time: lower decay rates extend protection but slow dynamics; higher rates accelerate diffusion but shorten the protected window. Maintaining uniform directional diffusion requires sufficient supercell count along the boundary (at least three supercells, with the central one flanked by two edge supercells). The approach is broadly applicable to other diffusion processes (e.g., ion transport), where ambient temperature and material parameters could actively tune topological versus ordinary states. Open questions include identifying a diffusion-system analog of magnetic-field-like quantities and fully establishing pseudospin analogies to spin-momentum locking known in QSHE wave systems.

The work introduces a QSHE-inspired, honeycomb-lattice thermal diffusion system that supports topologically protected edge modes capable of selecting the direction of macroscopic thermal diffusion. By exciting specific edge modes (A, A′, B), the system produces controllable thermal polarization along desired axes, with linear tunability via incoherent superposition. The edge modes exhibit robustness to defects and uniform, protected decay over a finite temporal window. These results expand topological control in diffusion from localization/decay to directional transport and polarization, suggesting new opportunities for thermal management and for extending topological strategies to other diffusion phenomena such as ion transport. Future research should optimize the decay-rate–duration trade-off, determine minimal device architectures (e.g., supercell counts) for robust operation, explore active tuning via external parameters, and clarify the diffusion analogs of pseudospin/magnetic-field quantities.

• Temporal limitation: Topological protection persists only over a finite time due to inherent diffusion; edge modes collapse as temperatures relax toward ambient (≈20 s for the chosen parameters).
• Structural requirement: A minimum of three supercells along the boundary is needed; unit cells at outer edges cannot sustain one-directional heat balance due to broken periodicity.
• Modeling scope: Results are based on numerical simulations with idealized adiabatic boundaries and tuned effective diffusivities; experimental validation and environmental effects (e.g., convection/radiation) are not addressed here.
• Physical analogy gaps: A clear counterpart to magnetic-field-like quantities and full pseudospin mapping in diffusion systems remains to be identified.