Signatures of quantum integrability and nonintegrability in the spectral properties of finite Hamiltonian matrices

For a two-spin model which is (classically) integrable on a five-dimensiona l hypersurface in six-dimensional parameter space and for which level degen eracies occur exclusively (with one known exception) on four-dimensional ma nifolds embedded in the integrability hypersurface, we investigate the rela tions between symmetry, integrability, and the assignment of quantum number s to eigenstates. We calculate quantum invariants in the form of expectatio n values for selected operators and monitor their dependence on the Hamilto nian parameters along loops within, without, and across the integrability h ypersurface in parameter space. We find clear-cut signatures of integrabili ty and nonintegrability in the observed traces of quantum invariants evalua ted in finite-dimensional invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action op erators as constituent elements of the Hamiltonian.