Exact and broken symmetries in a hydrodynamical description of chiral spin states

Following the procedure recently proposed by Wiegmann, we improve his analysis and derive a complete effective Lagrangian describing long-wavelength fluctuations around hypothetical three-dimensional (3D) and 2D chiral states of Heisenberg spin systems. We study realizations of the previously proposed high symmetry group of the continuous theory [SO(3,1)×SU(2)×U(1) in 3D and SO(2,1)×U_c(1)×U_s(1) in 2D] and observe that, in general, Lorentz symmetry is broken already in the bare Lagrangian, although it could be restored after a renormalization. In contrast to the conjectures made by Wiegmann, an additional non-Abelian SU(2) gauge symmetry expected in 3D (and its Abelian counterpart in 2D) are, in fact destroyed due to a spontaneous parity violation in chiral ground states.