Nonlinear sigma model Lagrangian for superfluid He3-A (B)

It is shown that the effective Lagrangian for the dynamics of superfluid He 3 is described by the language of the SU(2) chiral nonlinear sigma model in a unified way for A and B phases. The starting Lagrangian is assumed to be the nonlinear Schrodinger type. Here the key concept is to write the order parameter in terms of the rotation matrix from the intrinsic states and re write it with the equivalent SU(2) matrix. The resultant effective Lagrangi an is thus transcribed to the nonlinear sigma model for which the field tak es the value on the SU(2) manifold. The superfluid velocity of Mermin-Ho ty pe is discussed in this representation and shown to be given by the topolog ical quantity of SU(2).