Application of an external magnetic field can suppress thermal and quantum vibrations, allowing the coupling exchange interactions between ions to dominate, allowing antiferromagnetic order to exist at higher temperatures (as in (CH_3)_4NMnCl_3 for example). But what then causes a shift to lower temperature in other compounds? Surely the thermal and quantum vibrations are still suppressed in this case, and yet the Neel temperature decreases with field.
Just a wild guess: lower Neel temperature means lower (absolute) values of exchange parameters. Exchange integrals are quite sensitive to the distance between interacting ions. So, perhaps, the magnetostriction comes to play?
Except that any phase transition is caused by external influence (in this case by magnetic field)
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