Saddle index properties, singular topology, and its relation to thermodynamic singularities for a ϕ⁴ mean-field model

We investigate the potential energy surface of a ϕ⁴ model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers α₊,α₀,α₋ with α₊+α₀+α₋=1, provided that the interaction strength μ is smaller than a critical value. The saddle index n_s is equal to α₀ and its distribution function has a maximum at n_s^max=1∕3. The density p(e) of stationary points with energy per particle e, as well as the Euler characteristic χ(e), are singular at a critical energy e_c(μ), if the external field H is zero. However, e_c(μ)≠υ_c(μ), where υ_c(μ) 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≠0. The average saddle index n̅ _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 e̅ (n_s) is given by n_s(e̅ )=1+4e̅ (for H=0) that vanishes at e̅ =−1∕4>υ₀, the ground state energy.