Shear Induced Rigidity in Athermal Materials: Supplementary Information

The meanfield equations for α > 1, and ∆0 < ∆c admit three lines of transitions which end at a critical point (Rc,∆c). These lines are defined by Rt(∆0), RDST (∆0), and Rm(∆0) in descending order of magnitude (Fig.1(a) in main text). The line Rt(∆0), marks the transition from a single solution, M(H = 0) = 0, for R > Rt(∆0) to multiple solutions for M(H = 0), whereas the line Rm(∆0) marks the transition from multiple solutions for M(H = 0) (R > Rm(∆0)) to a single stable solution M(H = 0) = 0. The transition at Rt(∆0) is continuous, whereas the transition at Rm(∆0) is accompanied by discontinuous changes of both order parameters at H = 0. Rt(∆0) and Rm(∆0) can be calculated analytically from the mean field equations and yields: Rt(∆0) = √ −∆0