Displacement correlations in disordered athermal networks

We derive exact results for correlations in the displacement fields $\{ \delta \vec{r} \} \equiv \{ \delta r_{\mu = x,y} \}$ in near-crystalline athermal systems in two dimensions. We analyze the displacement correlations produced by different types of microscopic disorder, and show that disorder at the microscopic scale gives rise to long-range correlations with a dependence on the system size $L$ given by $\langle \delta r_{\mu} \delta r_{\nu} \rangle \sim c_{\mu \nu}(r/L,\theta)$. In addition, we show that polydispersity in the constituent particle sizes and random bond disorder give rise to a logarithmic system size scaling, with $c_{\mu \nu}(\rho,\theta) \sim \text{const}_{\mu\nu} - \text{a}_{\mu\nu}(\theta)\log \rho + \text{b}_{\mu\nu}(\theta) \rho^{2}$ for $\rho~(=r/L) \to 0$. This scaling is different for the case of displacement correlations produced by random external forces at each vertex of the network, given by $c^{f}_{\mu \nu}(\rho,\theta) \sim \text{const}^{f}_{\mu \nu} -( \text{a}^{f}_{\mu \nu}(\theta) + \text{b}^{f}_{\mu \nu}(\theta) \log \rho ) \rho^2$. Additionally, we find that correlations produced by polydispersity and the correlations produced by disorder in bond stiffness differ in their symmetry properties. Finally, we also predict the displacement correlations for a model of polydispersed soft disks subject to external pinning forces, that involve two different types of microscopic disorder. We verify our theoretical predictions using numerical simulations of polydispersed soft disks with random spring contacts in two dimensions.