Observation of bulk quadrupole in topological heat transport

The study addresses whether quantized bulk quadrupole moments and associated higher-order topological phases (with gapped edge and in-gap corner states) can exist in thermal diffusion, a domain governed by non-Hermitian (skew-Hermitian) physics. Prior works on higher-order topological insulators (HOTIs), including the Benalcazar–Bernevig–Hughes (BBH) model, have realized quadrupole phases in Hermitian and non-Hermitian wave systems, but translating these concepts to heat transport has been challenging due to the lack of quantized quadrupole moments and undefined negative couplings in diffusion. The purpose of this work is to reveal and experimentally observe quadrupole topological phases in fluid heat transport by engineering Hermitian advections and non-Hermitian thermal couplings, thereby expanding band topology in thermal systems. The significance lies in demonstrating hierarchical topological states on both real and imaginary bands in a diffusive medium, opening new avenues for diffusive metamaterial engineering and multipolar topological physics.

The authors review the foundations of topological states in classical systems, emphasizing Hermiticity for real-valued spectra and orthogonal eigenstates, and discuss non-Hermitian phenomena such as parity-time symmetry, skin effects, and Weyl exceptional rings. HOTIs have been predicted and observed in various Hermitian and non-Hermitian platforms. The BBH model underpins quantized quadrupole insulators with positive and negative couplings, and non-Hermitian extensions predict higher-order transitions in real-valued bands. In thermal diffusion, recent work established that dissipation leads to skew-Hermitian effective Hamiltonians, enabling non-Hermitian topological insulating phases and Weyl exceptional rings. However, existing thermal approaches could not realize quadrupole phases due to absent bulk quadrupole moments and the challenge of implementing negative-like couplings. This work builds on these insights to engineer effective quadrupole moments in a diffusive thermal system.

Theory and model: The system is a 2D square lattice composed of four-site unit structures. Each site is a finite-volume element of fluid heat transfer coupled to neighbors through thermal exchange channels. The energy equation per site is: ∂T/∂t = (κ/ρc)∇²T + (Ωμ/R)(∇T)ᵀ + (h_intracell/ρc)ΔT − (h_intercell/ρc)ΔT, where ρ, c, κ are material parameters; Ωμ is the imposed local advection magnitude; R and θ are polar coordinates; h denote heat transfer coefficients; and Q (s⁻¹) is the coupling strength. Intercell/intracell coupling ratio β controls relative coupling strengths. Advection at each site is decomposed into x and y components (Ω cosθ, Ω sinθ), enabling an effective two-component description. A wave-like temperature solution T = A exp[i(kx x − ωt + φx + ky y + φy)] captures oscillatory field propagation due to advection; the complex frequency includes a real part (from advection-induced momentum) and an imaginary part (from conduction and couplings), yielding complex bands. The four-site unit’s effective Hamiltonian (Supplementary Eq. 5) has real and imaginary parts associated with x and y components, permitting analysis of bulk, edge, and corner modes and dispersion in the Brillouin zone.

Experimental platforms: Two types of samples are fabricated from epoxy resin and immersed in water to form the fluid environment.

• Geometry: Square lattice with center-to-center spacing a = 56 mm. Each site is a hollow “advective ball” of radius 25 mm. Systems include lattices of 16 sites (unit demonstrations) and larger arrays (e.g., 9 square-lattices) totaling 144 sites and 264 coupling channels. Walls of balls and channels are 1 mm thick.
• Actuation: Each advective ball contains a steering gear set (bevel gears, 1:1) driven by motors via shafts along z to impose controlled angular velocities with specified in-plane advection directions.
• Materials and fluid: Water (thermal conductivity 0.6 W·m⁻¹·K⁻¹) and epoxy (also 0.6 W·m⁻¹·K⁻¹; ρ = 1180 kg·m⁻³, c = 750 J·kg⁻¹·m⁻³) minimize parasitic heat transfer.
• Measurements: Temperatures captured by IR camera (emissivity 0.97) and thermocouples. Field intensity defined via normalized block-averaged temperatures T′/⟨T⟩.

Two implementation strategies:

1. Hermitian advection route (β = 1): All coupling channels are identical to enforce equal intercell/intracell strengths. Advections are modulated to open real-valued bandgaps. For theory/estimation: Convective heat transfer coefficients yield Q ≈ 0.129 s⁻¹; example angular velocities: Ω1 = 1.3Q = 0.0205 rad·s⁻¹, Ω2 = 1.8Q = 0.0283 rad·s⁻¹, Ω3 = 2.7Q = 0.0424 rad·s⁻¹, Ω4 = 4.1Q = 0.0647 rad·s⁻¹. Experiments set Ωx = 1.3Q, sweep Ωy over [−3.154Ω, 0] to probe topological regimes. Steady-state measurements taken after ~30 minutes.
2. Non-Hermitian coupling route (0 < β < 1): All sites receive the same small advection to avoid modifying bands (Ω1 = Ω2 = 0.025Q = 0.0004 rad·s⁻¹). Intercell channels kept as in the advection case; intracell channels are enlarged and outfitted with internal fins to increase heat exchange area, tuning β. Cases implemented: β = 0.5 (enlarged channel with one fin), β = 0.2 and 0.1 (further added fins to achieve 5× and 10× intercell heat exchange areas, respectively). Longer stabilization times (~50–60 minutes) due to weaker advective effects.

Analysis: Band structures and dispersion relations are computed for real and imaginary parts of the eigenvalues in the first Brillouin zone. Wannier bands and nested Wilson loops along x and y yield quadrupole invariants and polarizations supporting topological character. Experiments probe bulk, edge, and corner responses at parameter values predicted to yield trivial/gapped/in-gap states.

• Existence and observation of quadrupole moments in heat transport: The work demonstrates that fluid heat transport can exhibit a quantized bulk quadrupole moment and higher-order topological phases in a non-Hermitian setting.
• Hierarchical states on complex bands: In stark contrast to classical wave systems, hierarchical bulk, gapped edge, and in-gap corner states are observed on both real-valued and imaginary-valued bands.
• Hermitian advection route (β = 1): By modulating advection alone, the real bands undergo topological transitions. Dispersion shows in-gap corner states and gapped edge states (e.g., for Ωx = −1.385Ω, Ωy = −2.077Ω), while Ωx = −3.154Ω yields a trivial insulating band structure. Experiments on a 9-lattice sample reveal two field-intensity peaks for bulk and edge states as Ωy is varied, and four in-gap corner states emerging near Ωy ≈ 3.05Q. A gapless imaginary band accompanies discrete real-band features.
• Non-Hermitian coupling route (0 < β < 1): With uniform small advection and unequal intercell/intracell couplings, gaps open in imaginary-valued bands while the real band remains gapless. Varying β from 1 to lower values induces two gaps (between bands 1–2 and 3–4), producing gapped edge and in-gap corner states. Experiments with β = 0.5, 0.2, 0.1 show: (i) two peaks in field intensity for bulk/edge and a single peak for corner responses; (ii) trivial bulk localization as β approaches 0 or 1 (imaginary-band localization). These features match numerical dispersion and temperature maps.
• Topological invariants: Both strategies exhibit gapped Wannier bands and half-integer quantized polarizations along x and y, with a nontrivial quadrupole invariant (value 1), confirming higher-order topology.
• Dynamics: Hierarchical states manifest during non-equilibrium transients before steady state, as captured by time-varying field-intensity rates.

The findings resolve whether higher-order topological phases—specifically quadrupole phases—can be realized in a purely diffusive thermal system lacking conventional negative couplings. By engineering Hermitian advections and non-Hermitian coupling asymmetries, the authors establish a quantized quadrupole moment and observe corner and edge states in heat transport. Crucially, complex eigenvalues enable topological phase transitions on both real and imaginary bands, extending higher-order topology beyond Hermitian wave analogs. This advances diffusive metamaterial engineering, offering robust, hierarchical heat-flow localization at corners and edges with potential for reconfigurability via advection or coupling design. The demonstration suggests broader relevance across diffusive domains and provides a platform to explore multipolar topological physics in complex-band settings.

This work creates an effective quadrupole moment in thermal diffusion and experimentally observes non-Hermitian quadrupole topological phases using two routes: (1) Hermitian advection with equal couplings (β = 1), and (2) non-Hermitian coupling asymmetry (0 < β < 1) with uniform advection. Both approaches yield hierarchical corner and edge states, validated by dispersion, experiments, and topological invariants (gapped Wannier bands, half-integer polarizations, quadrupole invariant 1). The results establish a new paradigm for topological heat transport with phase transitions on real and imaginary bands. Future directions include extending to fractal lattices and moiré superlattices, exploiting topological diffusion for controlling mass concentration in biomedicine and catalysis, and managing charge diffusion in semiconductors and other diffusive platforms.

The demonstrations are proof-of-concept on finite-sized, fabricated lattices (e.g., 16-site units and 9 square-lattices) immersed in water and made from epoxy, with specific parameter regimes (e.g., β tuning via intracell fins; advection magnitudes tied to Q). Steady-state measurements require relatively long times (tens of minutes). These factors may limit immediate generalizability and scalability, though the underlying principles are broadly applicable.