Since decades, rare-earth hexaborides are famous for their anomalously low work functions, making
them ideal materials for electron-emitting cathodes. Nevertheless, fundamental understanding
of their many unusual physical properties still remains hazy. The heavy-fermion metal CeB$_6$ stands out as the only material in this family that develops an exotic antiferroquadrupolar (AFQ) order at temperatures between 2.3 and 3.3 K, whose microscopic origin is still debated. Our inelastic-neutron-scattering data suggest the existence of an unusual spin resonance mode that emerges at the AFQ wavevector in the antiferromagnetic ground state of CeB$_6$ out of a broad continuum of itinerant paramagnetic fluctuations. The weakly dispersing tails of the main peak extend in momentum space, forming a rich anisotropic three-dimensional structure. The resonant mode is
sharply peaked in energy around 0.5 meV, below the onset of the charge gap, and is followed by a weaker second peak around 1.1 meV at the same wavevector. At lower energies, a suppression of the magnetic spectral weight marks the opening of a spin gap. The temperature evolution of this new mode bears striking resemblance to the spin resonance in unconventional superconductors, which could turn out to be a key to the understanding of the mysterious AFQ phase in CeB$_6$.
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