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Journal of Physics A Mathematical and General

Calculation of crossover exponent from Heisenberg to Ising behaviour using the fourth-order ε expansion

J E Kirkham

Published Nov 1, 1981• 16 citations
Publisher

Abstract

Extends the epsilon expansion for the crossover exponent phi from Heisenberg to Ising behaviour to fourth order in epsilon =4-d. Resummation of this expansion incorporating the high-order behaviour gives numerical estimates in three dimensions which are in agreement with high-temperature series expansion and experimental results.
DOI
10.1088/0305-4470/14/11/004

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The critical exponents $\ensuremath{\gamma}$, $\ensuremath{\eta}$ and the "crossover index" $\ensuremath{\phi}$ are computed for generalized classical Heisenberg models with nnn internal degrees of freedom as an exact expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ (ddd is the number of space dimensions). Results are obtained to order ${\ensuremath{\epsilon}}^{2}$ for $\ensuremath{\gamma}$ and to order ${\ensuremath{\epsilon}}^{3}$ for $\ensuremath{\eta}$. The results to this order for the three-dimensional Ising case ($n=\ensuremath{\epsilon}=1$) are $\ensuremath{\gamma}=1.244$ and $\ensuremath{\eta}=0.037$.
The tricirical exponents ${\ensuremath{\eta}}_{t}$ and ${\ensuremath{\varphi}}_{t}$ are calculated through order ${\ensuremath{\epsilon}}^{3}$ and ${\ensuremath{\epsilon}}^{2}$, respectively, in the isotropic nnn-component model, where $\ensuremath{\epsilon}=3\ensuremath{-}d$. The estimate for ${\ensuremath{\eta}}_{t}$, in two dimensions for the Ising case, is 0.027; the series for ${\ensuremath{\varphi}}_{t}$ is quite ill behaved, producing a negative estimate at this order. Beginning at this order in $\ensuremath{\epsilon}$, the spherical-model limit fails to exist. A scaling function for the spin-spin correlation function, appropriate for nonexceptional paths of approach to the tricritical point in the disordered phase, is calculated through ${\ensuremath{\epsilon}}^{2}$; its large momentum expansion is shown to involve the crossover exponent. These results are generalized to points where $\ensuremath{\theta}$ coexisting phases become critical.