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Stability of a half-quantum vortex (HQV) in superfluid $^3$He has beendiscussed recently by Kawakami, Tsutsumi and Machida in Phys. Rev. B {\bf 79},092506 (2009). We further extend this work here and consider the A$_2$ phase ofsuperfluid $^3$He confined in thin slab geometry and analyze the HQV realizedin this setting. Solutions of HQV and singly quantized singular vortex areevaluated numerically by solving the Ginzburg-Landau (GL) equation andrespective first critical angular velocities are obtained by employing thesesolutions. We show that the HQV in the A$_2$ phase is stable near the boundarybetween the A$_2$ and A$_1$ phases. It is found that temperature and magneticfield must be fixed first in the stable region and subsequently the angularvelocity of the system should be increased from zero to a sufficiently largevalue to create a HQV with sufficiently large probability. A HQV does not formif the system starts with a fixed angular velocity and subsequently thetemperature is lowered down to the A$_2$ phase. It is estimated that theexternal magnetic field with strength on the order of 1 T is required to have asufficiently large domain in the temperature-magnetic field phase diagram tohave a stable HQV.

Half-Quantum Vortices in Thin Film of Superfluid $^3$He