  The heavy fermion material CeB6 shows hidden magnetic ordered phases. Besides
well-known Ferromagnetic(FM) and Antiferromagnetic(AFM) phases, CeB6 is
speculated to form a unique antiferroquadrupolar (AFQ) phase that is orbital in
nature. Hidden from many characterization methods that cannot assess orbital
ordering, debate continue on the origins. From electronic-structure
calculations, we find that these three phases are energetically almost
degenerate, suggesting that magnetic domain walls form, possibly with defect
boundaries. Only calculations with spin-orbit coupling reproduce most band
structures and Fermi surfaces, as found in experiment, indicating the
importance of crystal-field splitting. Simulated ionization peaks, i.e., Ce f0
and f1 states, also agree with photoemission data. Small pressures stabilizes
the AFM over the FM phase, which is the observed phase at low temperature. We
propose to realize such small physical pressure under various situations e.g.
effect of intrinsic defects such as vacancies, antisites, surface effects etc.
