Characteristics of Fractons: from Specific Realizations to Ensemble Averages

Specific realizations due to computer experiments show that fracton excitations possess a core in which the vibrational amplitudes fall off sharply at their edges. When constructing the ensemble average of the fracton wave functions, we find \(\langle\phi_{\text{fr}}\rangle{\sim}\exp[-(r/\varLambda(\omega))^{d_{\phi}}]\), where \(\varLambda(\omega){\sim}\omega^{-0.71}\). The exponent, -0.71, is in close agreement with the prediction of the fracton dispersion law for a two-dimensional percolating network, -0.705, and the geometrical exponent d _φ is much larger than current theoretical limits for impurity electronic states. This suggests that the two problems are different.