The number of solutions of the Thouless-Anderson-Palmer equations forp-spin-interaction spin glasses

The number 〈Nₛ〉 of solutions of the equations of Thouless, Anderson, and Palmer for p-spin-interaction spin glass models is calculated. Below a critical temperature Tc this number becomes exponentially large, as it is in the Sherrington-Kirkpatrick model (p=2). But in contrast to this, for any p>2 the factor {α}(T)=N^-1ln〈Nₛ〉 jumps discontinuously at Tc(p), which is consistent with the discontinuity occurring within the mean-field theory for these models. For zero temperature the results obtained by Gross and M\'ezard are reproduced, and for p{→}{∞} one gets the result for the random energy model.