Statistical mechanics of one-dimensional Ginzburg-Landau fields. II. A test of the screening approximation n−1 expansion)

Abstract

The free-energy density of a one-dimensional Ginzburg-Landau field with $n$ components ${\ensuremath{\varphi}}_{3}$ is identified with the quantum-mechanical ground-state energy of an $n$-dimensional anharmonic oscillator. Normalizing the anharmonic term by ${(4n)}^{\ensuremath{-}1}$ and expanding in powers of ${n}^{\ensuremath{-}1}$, we find for the ground-state energy, $n{E}_{\ensuremath{\infty}}+{E}^{(1)}+{n}^{\ensuremath{-}1}{E}^{(2)}$, where ${E}_{\ensuremath{\infty}}=\frac{\ensuremath{\kappa}}{2}\ensuremath{-}\frac{1}{16{\ensuremath{\kappa}}^{2}}$, and ${E}^{(1)}=\frac{({\ensuremath{\kappa}}_{2}\ensuremath{-}2\ensuremath{\kappa})}{2}$ and ${E}^{(2)}=3{\ensuremath{\kappa}}_{2}^{\ensuremath{-}2}\ensuremath{-}6\ensuremath{\kappa}{\ensuremath{\kappa}}_{2}^{\ensuremath{-}3}\ensuremath{-}(\frac{11}{4}){\ensuremath{\kappa}}^{\ensuremath{-}1}{\ensuremath{\kappa}}_{2}^{\ensuremath{-}4}$ are the Hartree and first and second screening approximations, respectively. $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{2}={(4{\ensuremath{\kappa}}^{2}+{\ensuremath{\kappa}}^{\ensuremath{-}1})}^{\frac{1}{2}}$ are the inverse correlation lengths for ${\ensuremath{\varphi}}_{i}$ and $\ensuremath{\Sigma}{\ensuremath{\varphi}}_{i}^{2}$ respectively. The temperature is linearly related to the spring constant $\ensuremath{\tau}$, which in turn is connected with the correlation lengths by $\ensuremath{\tau}={\ensuremath{\kappa}}^{2}\ensuremath{-}\frac{1}{2\ensuremath{\kappa}}$. The analytic results are compared with exact numerical computations for $n=1, 2, 3, \mathrm{and} 4$. The second screening correction modestly improves the accuracy of the approximation.