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We study spatial spin and density self-ordering of a two-componentBose-Einstein condensate via collective Raman scattering into a linear cavitymode. The onset of the Dicke superradiance phase transition is marked by asimultaneous appearance of a crystalline density order and a spin-wave order.The latter spontaneously breaks the discrete $\mathbf{Z}_2$ symmetry betweeneven and odd sites of the cavity optical potential. Moreover, in thesuperradiant state the continuous $U(1)$ symmetry of the relative phase of thetwo condensate wavefunctions is explicitly broken by the cavity-inducedposition-dependent Raman coupling with a zero spatial average. Thus, thespatially-averaged relative condensate phase is locked at either $\pi/2$ or$-\pi/2$. This continuous symmetry breaking and relative condensate phaselocking by a zero-average Raman field can be considered as a genericorder-by-disorder process similar to the random-field-induced order in thetwo-dimensional classical ferromagnetic $XY$ spin model. However, the seed ofthe random field in our model stems from quantum fluctuations in the cavityfield and is a dynamical entity affected by self-ordering. The spectra ofelementary excitations exhibit the typical mode softening at the superradiancethreshold.

Disorder-Driven Density and Spin Self-Ordering of a Bose-Einstein Condensate in a Cavity