6533b7d6fe1ef96bd1266dfb

RESEARCH PRODUCT

Saddle index properties, singular topology, and its relation to thermodynamic singularities for aϕ4mean-field model

subject

description

We investigate the potential energy surface of a ${\ensuremath{\phi}}^{4}$ model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers ${\ensuremath{\alpha}}_{+},{\ensuremath{\alpha}}_{0},{\ensuremath{\alpha}}_{\ensuremath{-}}$ with ${\ensuremath{\alpha}}_{+}+{\ensuremath{\alpha}}_{0}+{\ensuremath{\alpha}}_{\ensuremath{-}}=1$, provided that the interaction strength $\ensuremath{\mu}$ is smaller than a critical value. The saddle index ${n}_{s}$ is equal to ${\ensuremath{\alpha}}_{0}$ and its distribution function has a maximum at ${n}_{s}^{\mathrm{max}}=1∕3$. The density $p(e)$ of stationary points with energy per particle $e$, as well as the Euler characteristic $\ensuremath{\chi}(e)$, are singular at a critical energy ${e}_{c}(\ensuremath{\mu})$, if the external field $H$ is zero. However, ${e}_{c}(\ensuremath{\mu})\ensuremath{\ne}{\ensuremath{\upsilon}}_{c}(\ensuremath{\mu})$, where ${\ensuremath{\upsilon}}_{c}(\ensuremath{\mu})$ 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 $H\ensuremath{\ne}0$. 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 $\overline{e}({n}_{s})$ is given by ${n}_{s}(\overline{e})=1+4\overline{e}$ (for $H=0$) that vanishes at $\overline{e}=\ensuremath{-}1∕4g{\ensuremath{\upsilon}}_{0}$, the ground state energy.