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

Abstract

We propose a one-dimensional (1D) diffusion equation (heat equation) for systems in which the diffusion constant (thermal diffusivity) varies alternately with a spatial period $a$. We solve the time evolution of the field (temperature) profile from a given initial distribution, by diagonalising the Hamiltonian, i.e., the Laplacian with alternating diffusion constants, and expanding the temperature profile by its eigenstates. We show that there are basically phases with or without edge states. The edge states affect the heat conduction around heat baths. In particular, rapid heat transfer to heat baths would be observed in a short time regime, which is estimated to be $t<10^{-2}$s for $a\sim 10^{-3}$m system and $t<1$s for $a\sim 10^{-2}$m system composed of two kinds of familiar metals such as titanium, zirconium and aluminium, gold, etc. We also discuss the effective lattice model which simplifies the calculation of edge states up to high energy. It is suggested that these high energy edge states also contribute to very rapid heat conduction in a very short time regime.

10 pages, 12 figures, v2: references added, text revised

References in corpus (4)

• Topological Field Theory of Time-Reversal Invariant Insulators
• Analogs of quantum Hall effect edge states in photonic crystals
• Observation of phononic helical edge states in a mechanical 'topological insulator'
• Topological properties of linear circuit lattices

Cited by in corpus (6)

• Localized heat diffusion in topological thermal materials
• Bulk-edge correspondence for nonlinear eigenvalue problems
• Higher-order topological heat conduction on a lattice for detection of corner states
• A symmetry-protected exceptional ring in a photonic crystal with negative index media
• Anomalous bulk-edge correspondence of nonlinear Rice-Mele model
• Fractional Thouless pumping of solitons: a unique manifestation of bulk-edge correspondence of nonlinear eigenvalue problems