Fig. 4: Shear anisotropy induced by hierarchical QLMs.
a Angular dependence of the simple shear modulus G(θ) in polar coordinates for the system shown in Fig. 2a. Five measurements are shown, from inner to outer: (light blue) unpinned system; (red) with key-core string atoms pinned (yellow atoms in b2); (yellow, nearly overlapped with red) with only key-core square atoms pinned; (blue) with both key-core atoms and θ-dependent strings pinned (yellow and cyan atoms in c2, θ = 90∘); (orange) with key-core atoms and all hidden strings pinned (magenta atoms in d). b1, c1 Excited atoms (cyan) with large non-affine displacements at θ = 45∘ (b1) and 90∘ (c1) when no pinning is applied. b2, c2 Same configurations after pinning the key-core atoms (yellow), showing suppression of the primary QLM at θ = 90∘ (c2) but not at θ = 45∘ (b2). d Spatial distribution of reduced transverse vibrational density of states, \({D}_{i}^{{{\rm{T}}}}(\omega )/\omega\) for ω = 0.8 ± 0.4. Magenta atoms represent the union of the key-core and θ-dependent string excitations. e Length distribution of θ-dependent strings with Lstrings > 3, showing a rapid decay in string length. f Angular dependence of the number of atoms participating in the key-core and hidden strings. This curve maps closely onto the anisotropy observed in G(θ) (compare with blue curve in a). g Frequency of pinning occurrences for each magenta atom from (d), as the system is rotated in 1∘ increments over 360∘. Atoms with Nangle = 360 (horizontal red dashed line) are the key-core atoms that are always excited. Other atoms show orientation-specific excitation, confirming the directional nature of string-like defects.
