Ground states for a class of deterministic spin models with glassy behaviour

We consider the deterministic model with glassy behaviour, recently introduced by Marinari, Parisi and Ritort, with Hamiltonian H = Sigma(i,j=1)(N) J(i,j)sigma(i) sigma(j), where J is the discrete sine Fourier transform. The ground state found by these authors for N odd and 2N+1 prime is shown to become asymptotically degenerate when 2N + 1 is a product of odd primes, and to disappear for N even. This last result is based on the explicit construction of a set of eigenvectors for J, obtained through its formal identity with the imaginary part of the propagator of the quantized unit symplectic matrix over the 2-torus.