Understanding thermal and phonon transport in solids has been of great importance in many disciplines such as thermoelectric materials, which usually requires an extremely low lattice thermal conductivity (LTC). By analyzing the finite-temperature structural and vibrational characteristics of typical thermoelectric compounds such as filled skutterudites and Cu 3 SbSe 3 , we demonstrate a concept of part-crystalline part-liquid state in the compounds with chemicalbond hierarchy, in which certain constituent species weakly bond to other part of the crystal. Such a material could intrinsically manifest the coexistence of rigid crystalline sublattices and other fluctuating noncrystalline sublattices with thermally induced largeamplitude vibrations and even flow of the group of species atoms, leading to atomic-level heterogeneity, mixed part-crystalline partliquid structure, and thus rattling-like thermal damping due to the collective soft-mode vibrations similar to the Boson peak in amorphous materials. The observed abnormal LTC close to the amorphous limit in these materials can only be described by an effective approach that approximately treats the rattling-like damping as a "resonant" phonon scattering.sublattice melting | partial Grüneisen parameter | first principles | anharmonicity U nderstanding thermal and phonon transport in solids has been of great importance in many disciplines such as thermoelectrics (1-3), phononic materials (4), and thermal management composites (5). The interplay among chemical bonds, lattice dynamics, and thermal transport in materials is also an attractive topic in condensed matter physics (6) and materials science (7). Thermal transport is a key issue in thermoelectric (TE) energy-conversion materials, which are regarded among the potential candidates for revolutionizing waste-heat recovery (2, 7-9). The dimensionless figure of merit of a TE material is defined as ZT = TS 2 σ=κ, where T, S, σ, and κ are the absolute temperature, Seebeck coefficient, electrical conductivity, and thermal conductivity, respectively. To improve the efficiency of TE conversion, many approaches aim at reducing the thermal conductivity, especially the lattice part, to a minimum level, namely the realization of phonon-glass-like thermal transport (1, 7).TE materials research primarily focuses on solid and crystalline thermoelectrics. It has been long viewed that all solids contain strong interatomic interactions without even an exception, and thus the established approaches to describe thermal transports in crystalline solids, including TE solids, are solely based on the perturbative "small-parameter" approximation to lattice dynamics of atoms around their equilibrium positions, i.e., phonons and phonon-phonon interactions (10, 11). As a result, crystallographic homogeneity at the atomic level in solid materials has overwhelmingly been accepted. However, recent work on exploring novel TE materials went noticeably beyond the conventional knowledge of solid TE compounds being ideally crystalline, atomically...