Influence of symmetry breaking on the fluctuation properties of spectra

We study the effect of gradual symmetry breaking in a non-integrable system on the level fluctuation statistics. We consider the case when the symmetry is represented by a quantum number that takes one of two possible values, so that the unperturbed system has a spectrum composed of two independent sequences. When symmetry-breaking perturbation is represented by a random matrix with an adjustable strength, the shape of the spectrum monotonically evolves towards the Wigner distribution as the strength parameter increases. This contradicts the observed behaviour of the acoustic resonance spectra in quartz blocks during the breaking of a point-group symmetry, which are changed in the beginning towards the Poisson statistics and then revert to the GOE statistics. This behaviour is explained by assuming that the symmetry-breaking perturbation removes the degeneracy due to symmetry for a limited number of levels, thus creating a third chaotic sequence. As symmetry breaking increases, the new sequence grows at the expense of the initial pair until it overwhelms the whole spectrum when the symmetry completely disappears. The calculated spacing distribution and spectral rigidity are able to describe the evolution of the acoustic resonance spectra.