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The quantum triangular XXZ model has recently enjoyed a wealth of newtheoretical results, especially in relation to the modeling of theBa$_3$CoSb$_2$O$_9$ compound. In particular, it has been understood that in alongitudinal magnetic field the umbrella (cone) phase, classically stable inall the easy-plane region of the ground-state phase diagram, is considerablyreduced by the effect of quantum fluctuations. We provide more quantitativeinformation for this phenomenon at arbitrary value of the site spin $S$, byemploying the dilute Bose gas expansion, valid in the high-field regime; ourresults improve the available estimates of the $1/S$ expansion. We quantify theextent to which a higher spin suppresses the effect of quantum fluctuations.Besides, we show how in three-dimensional layered systems a relatively smallantiferromagnetic interlayer coupling has a similar consequence of bringingback the umbrella phase in some part of the phase diagram.

Umbrella-coplanar transition in the triangular XXZ model with arbitrary spin