  Emergent crystals are periodic alignment of "emergent particles", i.e.,
localized collective behavior of atoms or their charges/spins/orbits. These
novel states of matter, widely observed in various systems, may deform under
mechanical forces with elasticity strikingly different from that of the
underlying material. However, their nonlinear and critical behaviors under
strong fields are hitherto unclear. Here we theoretically study the nonlinear
elasticity and structural transitions of skyrmion crystals (SkX) suffering
uniaxial distortion by using three different methods. Under moderate tension,
SkX behaves like a ductile material, with a negative crossover elastic
stiffness and a negative emergent Poisson's ratio at appropriate conditions of
magnetic field. Under strong straining, we observe at most six phase
transitions, leading to appearance of four novel emergent crystals that are
thermodynamically metastable. When subject to external loads, emergent crystals
rotate globally, and their composing "particles" have unlimited deformability,
which render their exotic polymorphism.
