Emergence of rigidity at the dynamic glass transition: a replica approach calculation

According to the mean-field replica theory of the glass transition, at the so-called dynamic transition the relaxation stops and the liquid freezes into one of many metastable states. We identify Goldstone modes of the resulting amorphous solid and derive a formal expression for its shear modulus. This expression is complementary to the formula used by Yoshino and Mezard [Phys. Rev. Lett. \textbf{105}, 015504 (2010)]. We combine our formal expression with the recently proposed version of the replica approach [G. Szamel, Europhys. Lett. \textbf{91}, 56004 (2010)] to calculate the shear modulus.