Two-level systems in a spin-glass model. I. General formalism and two-dimensional model

A spin-glass model with two-dimensional, isotropic spins (XY model) is studied. The model exhibits two-level systems (TLS) which are related to the sense of rotation of the spin direction. The two-dimensional version of the model is equivalent at low temperature to an Ising model with Coulomb interactions (which are logarithmic in two dimensions). However, this long-range effective interaction between TLS may be a special feature of two-dimensional spins. The model depends on a continuous parameter x and can reproduce, in particular, the 'Edwards-Anderson model', the 'Mattis model', and the 'odd model'.