Nematic ordering in pyrochlore antiferromagnets: high-field phase of chromium spinel oxides

Motivated by recent observation of a new high field phase near saturation in chromium spinels $A$Cr${}_2$O${}_4$ ($A=$ Zn, Cd, Hg), we study the $S=3\mathrm{/}2$ pyrochlore Heisenberg antiferromagnet with biquadratic interactions. Magnon instability analysis at the saturation field reveals that a very small biquadratic interaction can induce magnon pairing in pyrochlore antiferromagnets, which leads to the emergence of a ferro-quadrupolar phase, or equivalently a spin nematic phase, below the saturation field. We present the magnetic phase diagram in an applied field, studying both $S=3\mathrm{/}2$ and $S=1$ spin systems. The relevance of our result to chromium spinels is discussed.