Edge states of a diffusion equation in one dimension: Rapid heat conduction to the heat bath

Bulk-Edge Correspondence for Nonlinear Eigenvalue Problems

Although topological phenomena attract growing interest not only in linear systems but also in nonlinear systems, the bulk-edge correspondence under the nonlinearity of eigenvalues has not been established so far. We address this issue by introducing auxiliary eigenvalues. We reveal that the topological edge states of auxiliary eigenstates are topologically inherited as physical edge states when the nonlinearity is weak but finite (i.e., auxiliary eigenvalues are monotonic as for the physical one). This result leads to the bulk-edge correspondence with the nonlinearity of eigenvalues.

Higher-order topological heat conduction on a lattice for detection of corner states

A heat conduction equation on a lattice composed of nodes and bonds is formulated assuming the Fourier law and the energy conservation law. Based on this equation, we propose a higher-order topological heat conduction model on the breathing kagome lattice. We show that the temperature measurement at a corner node can detect the corner state which causes rapid heat conduction toward the heat bath, and that several-nodes measurement can determine the precise energy of the corner states.

A symmetry-protected exceptional ring in a photonic crystal with negative index media

Non-Hermitian topological band structures such as symmetry-protected exceptional rings (SPERs) can emerge for systems described by the generalized eigenvalue problem (GEVP) with Hermitian matrices. In this paper, we numerically analyze a photonic crystal with negative index media, which is described by the GEVP with Hermitian matrices. Our analysis using COMSOL Multiphysics® demonstrates that a SPER emerges for photonic crystals composed of split-ring resonators and metal-wire structures. We expect that the above SPER can be observed in experiments as it emerges at a finite frequency.

Geometric Phase and Localized Heat Diffusion

Many unusual wave phenomena in artificial structures are governed by their topological properties. However, the topology of diffusion remains almost unexplored. One reason is that diffusion is fundamentally different from wave propagation because of its purely dissipative nature. The other is that the diffusion field is mostly composed of modes that extend over wide ranges, making it difficult to be rendered within the tight-binding theory as commonly employed in wave physics. Here, the above challenges are overcome and systematic studies are performed on the topology of heat diffusion. Based on a continuum model, the band structure and geometric phase are analytically obtained without using the tight-binding approximation. A deterministic parameter is found to link the geometric phase with the edge state, thereby proving the bulk-boundary correspondence for heat diffusion. The topological edge state is experimentally demonstrated as localized heat diffusion and its dependence on the boundary conditions is verified. This approach is general, rigorous, and able to reveal rich knowledge about the system with great accuracy. The findings set up a solid foundation to explore the topology in novel thermal management applications.