Thouless-Anderson-Palmer Approach to the Spherical p-Spin Spin Glass Model

We analyze the Thouless-Anderson-Palmer (TAP) approach to the spherical p-spin spin glass model in zero external field. The TAP free energy is derived by summing up all the relevant diagrams for $N\to \infty$ of a diagrammatic expansion of the free energy. We find that if the multiplicity of the TAP solutions is taken into account, there is a first order transition in the order parameter at the critical temperature $T_{\rm c}$ higher than that predicted by the replica solution $T_{\rm RSB}$, but in agreement with the results of dynamics. The transition is of "geometrical" nature since the new state has larger free energy but occupies the largest volume in phase space. The transition predicted by the replica calculation is also of "geometrical" nature since it corresponds to the states with smallest free energy with positive complexity.