  We study the shear jamming of athermal frictionless soft spheres, and find
that in the thermodynamic limit, a shear-jammed state exists with different
elastic properties from the isotropically-jammed state. For example,
shear-jammed states can have a non-zero residual shear stress in the
thermodynamic limit that arises from long-range stress-stress correlations. As
a result, the ratio of the shear and bulk moduli, which in isotropically-jammed
systems vanishes as the jamming transition is approached from above, instead
approaches a constant. Despite these striking differences, we argue that in a
deeper sense, the shear jamming and isotropic jamming transitions actually have
the same symmetry, and that the differences can be fully understood by rotating
the six-dimensional basis of the elastic modulus tensor.
