We investigate the spin structure of many-fermion systems with a spin-conserving two-body random interaction. We find a strong dominance of spin-0 ground states and considerable correlations between energies and wave functions of low-lying states with different spin, but no indication of pairing. The spectral densities exhibit spin-dependent shapes and widths, and depend on the relative strengths of the spin-0 and spin-1 couplings in the two-body random matrix. The spin structure of low-lying states can largely be explained analytically.
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