Saddle index properties, singular topology, and its relation to thermodynamic singularities for a phi4 mean-field model.

We investigate the potential energy surface of a phi(4) model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers alpha(+) , alpha(0) , alpha(-) with alpha(+) + alpha(0) + alpha(-) =1 , provided that the interaction strength mu is smaller than a critical value. The saddle index n(s) is equal to alpha(0) and its distribution function has a maximum at n(max)(s) =1/3 . The density p (e) of stationary points with energy per particle e , as well as the Euler characteristic chi (e) , are singular at a critical energy e(c) (mu) , if the external field H is zero. However, e(c) (mu) not equal upsilon(c) (mu) , where upsilon(c) (mu) is the mean potential energy per particle at the thermodynamic phase transition point T(c) . This proves that previous claims that the topological and thermodynamic transition points coincide is not valid, in general. Both types of singularities disappear for H not equal 0 . The average saddle index n (s) as function of e decreases monotonically with e and vanishes at the ground state energy, only. In contrast, the saddle index n(s) as function of the average energy e ( n(s) ) is given by n(s) ( e ) =1+4 e (for H=0 ) that vanishes at e =-1/4> upsilon(0) , the ground state energy.