  We observe the joint spin-spatial (spinor) self-organization of a
two-component BEC strongly coupled to an optical cavity. This unusual
nonequilibrium Hepp-Lieb-Dicke phase transition is driven by an off-resonant
two-photon Raman transition formed from a classical pump field and the emergent
quantum dynamical cavity field. This mediates a spinor-spinor interaction that,
above a critical strength, simultaneously organizes opposite spinor states of
the BEC on opposite checkerboard configurations of an emergent 2D lattice. The
resulting spinor density-wave polariton condensate is observed by directly
detecting the atomic spin and momentum state and by holographically
reconstructing the phase of the emitted cavity field. The latter provides a
direct measure of the spin state, and a spin-spatial domain wall is observed.
The photon-mediated spin interactions demonstrated here may be engineered to
create dynamical gauge fields and quantum spin glasses.
