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Physical review. B, Solid state

Low-temperature renormalization group for ferromagnets with long-range interactions

Josef Sak

Published May 1, 1977• 79 citations
Publisher

Abstract

Ferromagnets with the number of spin components $n>2$ and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on $d$ and $n$ but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to $d$ and $n$.

DOI
10.1103/physrevb.15.4344

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Ferromagnets with the number of spin components n>2n>2n>2 and with power-law interactions r\ensuremath−d\ensuremath−\ensuremathσ{r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}r\ensuremath−d\ensuremath−\ensuremathσ are studied in dimensions d=2+\ensuremathϵd=2+\ensuremath{\epsilon}d=2+\ensuremathϵ for small \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. The following theorem is proved: Let \ensuremathηSR{\ensuremath{\eta}}_{\mathrm{SR}}\ensuremathηSR​ be the value of the critical exponent \ensuremathη\ensuremath{\eta}\ensuremathη for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality 2\ensuremath−\ensuremathσ<\ensuremathηSR2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}2\ensuremath−\ensuremathσ<\ensuremathηSR​ is satisfied. In this case the critical exponents depend on ddd and nnn but not on \ensuremathσ\ensuremath{\sigma}\ensuremathσ. In the opposite case, the value of \ensuremathη\ensuremath{\eta}\ensuremathη is \ensuremathηLR=2\ensuremath−\ensuremathσ{\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}\ensuremathηLR​=2\ensuremath−\ensuremathσ and the critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν depends on \ensuremathσ\ensuremath{\sigma}\ensuremathσ in addition to ddd and nnn.
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Critical exponents for a ddd-dimensional system with an isotropic nnn-component order parameter and long-range attractive interactions decaying as $\frac{1}{{r}^{d+\ensuremath{\sigma}}} (\ensuremath{\sigma}>0)$ are derived, using the renormalization group approach, as power series in $\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d>0$ ($\ensuremath{\sigma}\ensuremath{\ne}2$, fixed) or $\ensuremath{\Delta}\ensuremath{\sigma}=\ensuremath{\sigma}\ensuremath{-}\frac{1}{2}d>0$ (ddd fixed) and, separately, to order 1n\frac{1}{n}n1​ for all ddd and $\ensuremath{\sigma}\ensuremath{\ne}2$. For $\ensuremath{\epsilon}<0$ the exponents have fixed ("classical") values; when $\ensuremath{\epsilon}=\ensuremath{\Delta}\ensuremath{\sigma}=0$ fractional powers of $\mathrm{ln}(\frac{\ensuremath{\Delta}T}{{T}_{c}})$ appear; when $\ensuremath{\sigma}>2$ the exponents assume their short-range values.
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