Splitting of real squashing mode in acoustic-impedance experiments on B3

A general analysis is presented for various time-reversal-preserving symmetry-breaking fields that split the collective modes of B3. In the absence of these fields, the excited states of the order-parameter factor into pure-real and pure-imaginary components and these two categories have different frequencies (i.e., are nondegenerate). The symmetry analysis made in this paper avoids the usual analogies with the quantum-mechanical theory of angular momentum (which involve complex wave functions and hence do not preserve the real-imaginary separation of the order-parameter components). Our analysis is based on the interplay of real representations of the rotation groups SO(3) and SO(2). For the J=2+ mode we use the fact that the real, irreducible, five-dimensional representation of SO(3) can be resolved, due to the time-reversal-preserving symmetry breaking mentioned above, into the direct sum of two two-dimensional, and one one-dimensional (trivial) real representations of the two-dimensional rotation group SO(2): 5SO(3)=2SO(2)+2SO(2)+1SO(2). A similar decomposition is possible for the J=1+ mode; corresponding statements can be made for the pure-imaginary (-) modes.