Self-Consistent Screening Calculation of the Critical Exponentη

Abstract

A self-consistent version of the $\frac{1}{n}$ expansion is used to calculate the critical exponent $\eta \mathrm{}(n,d)\mathrm{}$ for an $n$-component Ginzburg-Landau field with spatial dimensionality $d$. The result is exact to first order in $\frac{1}{n}$ but also includes a partial summation of graphs to all orders in $\frac{1}{n}$. This leads to a bounding of $\eta$ for small $n$, in contrast to the simple $\frac{1}{n}$ expansion. Results are $\eta \mathrm{}(3,3)\mathrm{}\simeq 0.079$, $\eta \mathrm{}(2,3)\mathrm{}\simeq 0.11$, and $\eta \mathrm{}(1,3)\mathrm{}\simeq 0.177$. For $d=2\mathrm{}$ the theory leads to the conjecture that $\eta$ vanishes for large values of $n$.