  Solids are distinguished from fluids by their ability to resist shear. In
traditional solids, the resistance to shear is associated with the emergence of
broken translational symmetry as exhibited by a non-uniform density pattern,
which results from either minimizing the energy cost or maximizing the entropy
or both. In this work, we focus on a class of systems, where this paradigm is
challenged. We show that shear-driven jamming in dry granular materials is a
collective process controlled solely by the constraints of mechanical
equilibrium. We argue that these constraints lead to a broken translational
symmetry in a dual space that encodes the statistics of contact forces and the
topology of the contact network. The shear-jamming transition is marked by the
appearance of this broken symmetry. We extend our earlier work, by comparing
and contrasting real space measures of rheology with those obtained from the
dual space. We investigate the structure and behavior of the dual space as the
system evolves through the rigidity transition in two different shear
protocols. We analyze the robustness of the shear-jamming scenario with respect
to protocol and packing fraction, and demonstrate that it is possible to define
a protocol-independent order parameter in this dual space, which signals the
onset of rigidity.
