  Ripples in pristine freestanding graphene naturally orient themselves in an
array that is alternately curved-up and curved-down; maintaining an average
height of zero. Using scanning tunneling microscopy (STM) to apply a local
force, the graphene sheet will reversibly rise and fall in height until the
height reaches 60-70 percent of its maximum at which point a sudden, permanent
jump occurs. We successfully model the ripples as a spin-half Ising magnetic
system, where the height of the graphene is the spin. The permanent jump in
height, controlled by the tunneling current, is found to be equivalent to an
antiferromagnetic-to-ferromagnetic phase transition. The thermal load
underneath the STM tip alters the local tension and is identified as the
responsible mechanism for the phase transition. Four universal critical
exponents are measured from our STM data, and the model provides insight into
the statistical role of graphenes unusual negative thermal expansion
coefficient.
