Saddle index properties, singular topology, and its relation to thermodynamic singularities for a ${\varphi }^4$ mean-field model

We investigate the potential energy surface of a ${\varphi }^4$ 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 $n_s^{\max }=1∕3$. The density $p\left(e\right)$ of stationary points with energy per particle $e$, as well as the Euler characteristic $\chi \left(e\right)$, are singular at a critical energy $e_c\left(\mu \right)$, if the external field $H$ is zero. However, $e_c\left(\mu \right)\ne {\upsilon }_c\left(\mu \right)$, where ${\upsilon }_c\left(\mu \right)$ 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 $H\ne 0$. The average saddle index ${\overline{n}}_s$ 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 $\overline{e}\left(n_s\right)$ is given by $n_s\left(\overline{e}\right)=1+4\overline{e}$ (for $H=0$) that vanishes at $\overline{e}=-1∕4>{\upsilon }_0$, the ground state energy.

Figure 1

(a) ${\alpha }_{+}$ dependence of $\upsilon ({\alpha }_{+},{\alpha }_0)$ for $\mu =0.2$ and $H=0$; (b) same for $H=0.1$. Dotted lines locate the maxima. Note the concavity of $\upsilon ({\alpha }_{+},{\alpha }_0)$ in ${\alpha }_{+}$ for $H⩾0$ and its symmetry with respect to the maximum position ${\alpha }_{+}^{\max }=(1-{\alpha }_0)∕2$ for $H=0$.

Figure 2

(a) Configurational entropy $s(e,n_s)$ versus $n_s$ for different energies and zero field; (b) same for nonzero field.

Figure 3

(a) Averaged saddle index ${\overline{n}}_s\left(e\right)$ versus $e$ for $\mu =0.1$,0.2, $1∕3$ and $H=0$; (b) same for $\mu =0.2$ and $H=0$ and $H=0.15$.

Figure 4

$\mu$-dependence of the critical average saddle index $n_s^{\left(c\right)}\left(\mu \right)$ for $0⩽\mu ⩽1∕3$ (solid line). The dashed line is the asymptotic result Eq. (B5) for $\mu ⪡1$.

Figure 5

Saddle index $n_s\left(\overline{e}\right)$ versus averaged energy $\overline{e}$ for $\mu =0.2$ and $H=0$ and 0.15.

Figure 6

(a) Energy dependence of the configurational entropy $s\left(e\right)$ for different interaction strength $\mu$, with and without field. Dashed line is the solution for $\mu =0$ and $H=0$ and the circle indicates the location of $e_c\left(\mu \right)$; (b) Derivative of the configurational entropy showing a transition at $e=e_c\left(\mu \right)$ for $H=0$.