Glass phase of two-dimensional triangular elastic lattices with disorder

We study two-dimensional triangular elastic lattices in a background of point disorder, excluding dislocations (tethered network). Using both (replica symmetric) static and (equilibrium) dynamic renormalization group (RG) for the corresponding N=2 component model, we find a transition to a glass phase for T${\mathrm{T}}_{\mathrm{g}}$, described by a plane of perturbative fixed points. The growth of displacements is found to be asymptotically isotropic with ${\mathrm{u}}_{\mathrm{T}}^2$∼${\mathrm{u}}_{\mathrm{L}}^2$∼${\mathrm{A}}_1$ln ${}_{}{}^2{}_{}{}^{}\mathrm{r}$, with universal subdominant anisotropy ${\mathrm{u}}_{\mathrm{T}}^2$-${\mathrm{u}}_{\mathrm{L}}^2$∼${\mathrm{A}}_2$ln r, where ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ depend continuously on temperature and the Poisson ratio σ. We also obtain the continuously varying dynamical exponent z. For the Cardy-Ostlund N=1 model, a particular case of the above model, we point out a discrepancy in the value of ${\mathrm{A}}_1$ with other published results in the literature. We find that our result reconciles the order of magnitude of the RG predictions with the most recent numerical simulations.