Criticality in subisostatic spring networks stabilized by thermal fluctuations*

A variety of elastic structures can be modeled by interconnected networks of Hookean springs. As Maxwell showed, such spring networks with connectivity below a critical or isostatic threshold are unstable for small deformations. These subisostatic networks, however, can be stabilized by various mechanisms including adding interactions such as bending rigidity between adjacent bonds or springs. It is also possible to stabilize otherwise floppy spring networks by thermal fluctuations. Under nonlinear deformations, however, even subisostatic spring networks undergo a phase transition from a floppy to a rigid state. This transition is critical in nature and exhibits a variety of critical exponents near a critical strain that depends on network connectivity. Here, we explore the criticality aspects of this strain-induced transition in presence of thermal fluctuations.