Selection rules for the high harmonic generation spectra

For three-dimensional many-electron time periodic Hamiltonians which are invariant under dynamical symmetry of order N we prove that only the (nN±1)th, n = 1, 2,…, harmonics are generated. We discuss the application of the dynamical symmetry based selection rules to the generation of high harmonics by thin crystals. The derived selection rules are demonstrated numerically for a one-dimensional model, showing that the dynamically symmetric systems can be used not only as “filters” of the very high harmonics but also as their “amplifiers”.