EXACT AND BROKEN SYMMETRIES IN A HYDRODYNAMICAL DESCRIPTION OF CHIRALSPIN STATES

Following the procedure recently proposed by Wiegmann, we improve his analysis and derive a complete effective Lagrangian describing long-wa velength fluctuations around hypothetical three-dimensional 3D) and 2D chiral states of Heisenberg spin systems. We study realizations of th e previously proposed high symmetry group of the continuous theory [SO (3,1)xSU(2)xU(l) in 3D and SO(2,1)xU(c)(1)xU(s)(1) in 2D] and observe that, in general, Lorentz symmetry is broken already in the bare Lagra ngian, although it could be restored after a renormalization. In contr ast to the conjectures made by Wiegmann, an additional non-Abelian SU( 2) gauge symmetry expected in 3D (and its Abelian counterpart on 2D) a re, in fact, destroyed due to a spontaneous parity violation in chiral ground states.