A characteristic energy scale in glasses.

Glasses feature a broad distribution of relaxation times and activation energies without an obvious characteristic scale. At the same time, macroscopic quantities such as Newtonian viscosity and nonlinear plastic deformation, are interpreted in terms of a characteristic energy scale, e.g. an effective temperature-dependent activation energy in Arrhenius relations. Nevertheless, despite its fundamental importance, such a characteristic energy scale has not been robustly identified. Inspired by the accumulated evidence regarding the crucial role played by soft quasilocalized excitations in glassy dynamics, we propose that the bulk average of the glass response to a localized force dipole defines such a characteristic energy scale. We show that this characteristic glassy energy scale features remarkable properties: $(i)$ It increases dramatically with decreasing temperature of equilibrium supercooled states, significantly surpassing the corresponding increase in the shear modulus, dismissing the common view that structural variations in supercooled liquids upon vitrification are minute $(ii)$ Its variation with annealing and system size is very similar in magnitude and form to that of the energy of the softest non-phononic vibrational mode, thus establishing a very unusual relation between a rare glassy fluctuation and a bulk average $(iii)$ It exhibits striking dependence on spatial dimensionality and system size, due to the long-ranged fields associated with quasilocalization, which are speculated to be related to peculiarities of the glass transition in two dimensions. In addition, we identify a truly-static growing lengthscale associated with the characteristic glassy energy scale, and discuss possible connections between the increase of this energy scale and the slowing down of dynamics near the glass transition temperature. Open questions and future directions are discussed.