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Journal of Physics A Mathematical Nuclear and General

Statistical mechanics of one-dimensional Ginzburg-Landau fields: Feynman graph evaluation of the screening approximation (n-1expansion)

Alan J. Bray

Published Nov 21, 1974• 29 citations
Publisher

Abstract

The free energy per component of an isotropic one-dimensional Ginzburg-Landau field with n components is calculated to order n -2 by evaluating the appropriate Feynman graphs. The result agrees with that of Ferrell and Scalapino (1974) obtained by solving the equivalent anharmonic oscillator problem, but the method has greater generality.

DOI
10.1088/0305-4470/7/17/009

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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.

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The free energy per component of an isotropic one-dimensional Ginzburg-Landau field with n components is calculated to order n -2 by evaluating the appropriate Feynman graphs. The result agrees with that of Ferrell and Scalapino (1974) obtained by solving the equivalent anharmonic oscillator problem, but the method has greater generality.
The free-energy density of a one-dimensional Ginzburg-Landau field with nnn components ${\ensuremath{\varphi}}_{3}$ is identified with the quantum-mechanical ground-state energy of an nnn-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,and4n=1, 2, 3, \mathrm{and} 4n=1,2,3,and4. The second screening correction modestly improves the accuracy of the approximation.
We have been studying the statistical mechanics of one-dimensional Ginzburg-Landau fields for real, complex, and phase-only fields. Here, results for the specific heat, the order-parameter-order-parameter, and energy-density-energy-density correlation functions, will be presented. Formally, these solutions are of interest because they describe the behavior of systems which are nearly ordered but do not undergo sharp phase transitions. Physically, the real-field results may have application in some organic chain systems, while the complex field and phase-only fields are associated with superconducting strips and linear arrays of coupled weak links, respectively.
In an earlier paper the Ginzburg-Landau free energy functional was used to calculate the effect of thermodynamic fluctuations on the off-diagonal correlation function and we found no off-diagonal long-range order in one- and two-dimensional systems. It has been pointed out that, for a charged system, the use of the Ginzburg-Landau free energy functional is in error for arbitrary nonequilibrium values of the order parameter since the electrostatic energy of the charge fluctuations associated with an arbitrary order parameter is not included in the free energy functional. We have not succeeded in a direct generalization of the free energy functional so we are forced to proceed by inference from the generalized random phase approximation (RPA). We find that, for uncharged systems, the RPA gives a linearization of the results obtained earlier using the Ginzburg-Landau theory. For charged systems we find in the RPA results similar to those obtained for uncharged systems. From this we conclude that it is very likely that, as in uncharged systems, there will be no ODLRO in charged infinite one- and two-dimensional systems.
A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential V(x)V(x)V(x). For the oscillator potential V(x)=kx22V(x)=\frac{k{x}^{2}}{2}V(x)=2kx2​ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, $V(x)=\ensuremath{-}\frac{A{x}^{2}}{2}+\frac{B{x}^{4}}{4}$, is calculated over a range of the positive constants AAA and BBB and compared with numerical solutions of the Schr\"odinger equation.