Critical Behavior of Magnets with Dipolar Interactions. V. Uniaxial Magnets in $d$ Dimensions

The exact renormalization-group approach is used to study the critical behavior for

T > T_{c}

H = 0

of a uniaxial ferromagnetic (or ferroelectric) system in

d

dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for

t = \frac{T}{T_{c}} - 1 ≫ \hat{g} = \frac{{(g μ_{B})}^{2}}{J a^{d}}

, where

J

is the exchange parameter,

g μ_{B}

is the magnetic moment per spin, and

a

is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when

t^{φ} \approx \hat{g}

, where

φ = 1 + \frac{ε}{6}

(to first order in

ε = 4 - d

). For

t ≪ \hat{g}

, the leading temperature singularity in the Fourier transform of the spin-spin correlation function

Γ (\vec{q})

becomes

ξ^{2} \times {[1 + {(ξ q)}^{2} - h_{0} {(ξ q^{z})}^{2} + g_{0} {(\frac{q^{z}}{q})}^{2}]}^{- 1}

, where

h_{0}

and

g_{0}

are of order

\hat{g}

and

ξ (t)

varies as

t^{- \frac{1}{2}}

for

d > 3

, as

t^{- \frac{1}{2}} {| \ln t |}^{\frac{1}{6}}

for

d = 3

, and as

t^{- ν}

with

\frac{1}{2 ν} = 1 - \frac{(3 - d)}{6} + O ({(3 - d)}^{2})

for

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Critical Behavior of Magnets with Dipolar Interactions. V. Uniaxial Magnets indDimensions