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

The free-energy density of a one-dimensional Ginzburg-Landau field with

n

components

φ_{3}

is identified with the quantum-mechanical ground-state energy of an

n

-dimensional anharmonic oscillator. Normalizing the anharmonic term by

{(4 n)}^{- 1}

and expanding in powers of

n^{- 1}

, we find for the ground-state energy,

n E_{\infty} + E^{(1)} + n^{- 1} E^{(2)}

, where

E_{\infty} = \frac{κ}{2} - \frac{1}{16 κ^{2}}

, and

E^{(1)} = \frac{(κ_{2} - 2 κ)}{2}

and

E^{(2)} = 3 κ_{2}^{- 2} - 6 κ κ_{2}^{- 3} - (\frac{11}{4}) κ^{- 1} κ_{2}^{- 4}

are the Hartree and first and second screening approximations, respectively.

κ

and

κ_{2} = {(4 κ^{2} + κ^{- 1})}^{\frac{1}{2}}

are the inverse correlation lengths for

φ_{i}

and

Σ φ_{i}^{2}

respectively. The temperature is linearly related to the spring constant

τ

, which in turn is connected with the correlation lengths by

τ = κ^{2} - \frac{1}{2 κ}

. The analytic results are compared with exact numerical computations for

n = 1, 2, 3, and 4

. The second screening correction modestly improves the accuracy of the approximation.

This publication has 18 references indexed in Scilit:

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