Observation of bulk quadrupole in topological heat transport

Topological states of matter have seen significant advancements across various classical wave fields. In adiabatic systems, Hermiticity is crucial for topological properties, ensuring real-valued eigenvalues and orthogonal eigenstates. Open systems, however, introduce non-Hermiticity due to interactions with the environment. While dissipation violates the bulk-boundary correspondence in Hermitian systems, it enables exotic properties such as parity-time symmetry, skin effects, and Weyl exceptional rings. Higher-order topological insulators (HOTIs) have further expanded research into hierarchical features in both Hermitian and non-Hermitian systems. The Benalcazar-Bernevig-Hughes (BBH) model, characterized by a quantized bulk quadrupole moment, is key to realizing a minimal quadrupole topological insulator (QTI). A modified non-Hermitian BBH model suggests that both on-site non-Hermiticity and Hermiticity can lead to quadrupole topological phases. Recent findings show that dissipative diffusion is governed by skew-Hermitian physics, characterized by a purely imaginary Hamiltonian, leading to unique topological features in heat transport, including non-Hermitian topological insulating phases and Weyl exceptional rings. However, current methods fail to create non-Hermitian thermal quadrupole topological phases due to the absence of a bulk quadrupole moment and undefined negative couplings in heat transfer. This paper addresses this gap by demonstrating the existence of a quadrupole moment and non-Hermitian quadrupole topological phases in heat transport.

The study extensively reviews existing literature on topological states of matter, focusing on the contrast between Hermitian and non-Hermitian systems and the emergence of higher-order topological insulators (HOTIs). It highlights the significance of the Benalcazar-Bernevig-Hughes (BBH) model and its modified non-Hermitian counterpart in understanding quadrupole topological insulators (QTIs). The review also covers recent advancements in understanding dissipative diffusion and its connection to skew-Hermitian physics, emphasizing the challenges in achieving non-Hermitian thermal quadrupole topological phases due to the absence of a bulk quadrupole moment in existing thermal transport models.

The researchers designed a convective fluid heat transport system with multiple discrete sites, where each site represents a finite volume of heat transfer. Tunable advections were introduced on each site to create effective oscillations and form an effective unit structure of four neighboring sites, periodically configured to establish a 2D square lattice. The general heat energy equation for each site was expressed, considering density, specific heat, thermal conductivity, angular velocities of convection, radial and azimuth components, and heat transfer coefficients. The continuous conditions for quantization were considered. Tunable advections were used to create effective oscillations and a unit structure consisting of four neighboring sites. These structures were periodically configured to create a 2D square lattice. The heat energy equation was utilized to model the system, with advections acting as real Hermiticity, equivalent to gain and loss in photonics. The tilted connections between adjacent sites led to varying isotherm orientations and coupling degrees. This enabled over-coupling and under-coupling relative to a reference coupling strength. A wave-like solution was adopted to describe the oscillatory temperature field propagations, leading to an effective Hamiltonian for the four-site unit structure. The complex angular frequency and eigenvalues indicated complex bands, with the imaginary part originating from intrinsic conduction and thermal couplings, and the real part representing the effective momentum from advections. Two strategies were used to demonstrate quadrupole topological phases: modulating Hermitian advection and non-Hermitian coupling. A square lattice with 16 sites was fabricated, immersed in water, with hollow sites and tailored thermal coupling strengths. For the Hermitian advection case, specific advections were implemented to ensure real eigenvalues. The dispersion relations were calculated to validate the existence of quadrupole topological phases. For the non-Hermitian coupling case, the intracell and intercell thermal coupling strengths were manipulated by altering heat exchange areas, introducing internal fins to enhance intracell couplings. The imaginary band structures were analyzed to identify topological states. Experiments involved measuring temperature distributions using an IR camera and thermocouples to observe the hierarchical features at the bulk, edge, and corner of the fabricated samples. The thermal profiles were captured after the temperature distributions reached steady state.

The study successfully generated a quantized bulk quadrupole moment in fluid heat transport and experimentally observed non-Hermitian thermal quadrupole topological phases. The key findings include: 1. **Hierarchical States in Real and Imaginary Bands:** The experiments demonstrated the existence of bulk, gapped edge, and in-gap corner states in both the real and imaginary parts of the complex band structure. This is a significant departure from previously observed higher-order states, which were only found in real-valued bands in classical wave fields. 2. **Quadrupole Topological Phases Induced by Hermitian Advection:** Modulating the Hermitian advection created quadrupole topological phases. Experimental results showed localized temperature distributions at the corner, edge, and bulk, confirming the theoretical predictions. 3. **Quadrupole Topological Phases Induced by Non-Hermitian Coupling:** Adjusting the non-Hermitian thermal coupling strengths also generated quadrupole topological phases, observable along the imaginary-valued bands. Experiments confirmed the presence of in-gap corner states, gapped edge states, and trivial bulk states by manipulating the ratio between intercell and intracell thermal coupling strengths (β). 4. **Topological Robustness:** The topological robustness of the observed phases was confirmed through the calculation of nontrivial quadrupole invariants and half-integer polarizations. 5. **Significance of Complex Eigenvalues:** The use of complex eigenvalues enabled the observation of topological phase transitions in both real and imaginary-valued bands, highlighting the unique properties of higher-order diffusive quadrupoles. 6. **Experimental Validation:** The study provides robust experimental validation through the fabrication of thermal systems and precise control of advections and thermal coupling strengths. The measured temperature distributions strongly support the theoretical predictions and numerical simulations. 7. **Potential Applications:** The results open up exciting possibilities for controlling mass concentration in biomedicine and catalysis, charge diffusion in semiconductors, and other diffusive phenomena.

The observed non-Hermitian thermal quadrupole topological phases significantly deviate from previously understood higher-order topological insulators (HOTIs) in classical wave fields. The ability to observe these phases in both real and imaginary bands, achieved by manipulating either Hermitian advection or non-Hermitian thermal coupling, is a major contribution. The findings demonstrate a new avenue for understanding and controlling topological diffusion, potentially impacting diverse fields such as biomedicine, catalysis, and semiconductor physics. The robust experimental validation and theoretical modeling provide a strong foundation for future research exploring topological phenomena in complex systems.

This study successfully demonstrated the creation of quantized bulk quadrupole moments in heat transport and the observation of non-Hermitian thermal quadrupole topological phases. The ability to induce these phases by manipulating either Hermitian advection or non-Hermitian thermal coupling is a significant advancement. Future research could explore topological diffusion in more complex systems, such as fractal systems and moiré lattices, and investigate potential applications in various fields.

While the study provides compelling evidence for non-Hermitian thermal quadrupole topological phases, there are limitations. The experimental setup requires precise control of advections and thermal couplings, which may be challenging to achieve in more complex or less controlled environments. The system's size and geometry may also affect the generalizability of the results. Future work could focus on scaling the system and exploring the robustness of the topological phases under different conditions.