We study the quantum spin-
21 Heisenberg model in two dimensions, interacting through a nearest-neighbor antiferromagnetic exchange
(J) and a ferromagnetic dipolarlike interaction
(Jd), using double-time Green's function, decoupled within the random phase approximation. We obtain the dependence of
kBTc∕Jd as a function of frustration parameter $\ensuremath{\delta}$, where
Tc is the ferromagnetic (F) transition temperature and $\ensuremath{\delta}$ is the ratio between the strengths of the exchange and dipolar interaction (i.e., $\ensuremath{\delta}=J∕{J}_{d}$). The transition temperature between the F and paramagnetic phases decreases with $\ensuremath{\delta}$, as expected, but goes to zero at a finite value of this parameter, namely, $\ensuremath{\delta}={\ensuremath{\delta}}_{c}=\ensuremath{\pi}∕8$. At
T=0 (quantum phase transition), we analyze the critical parameter ${\ensuremath{\delta}}_{c}(p)$ for the general case of an exchange interaction in the form
Jij=Jd∕rijp, where ferromagnetic and antiferromagnetic phases are present.
The idea of the spectrum generating algebra (SGA) is used to study the low-lying excitations in various localised-spin models of ferromagnetic, ferrimagnetic and antiferromagnetic solids. The noncompact algebra SO(2,1) approximately SU(1,1) is found to generate the spin wave or the magnon spectrum for ferromagnetic, ferrimagnetic and antiferromagnetic Bravais lattices with nearest-neighbour exchange interactions (Heisenberg model). The ferromagnetic case is treated including dipolar interactions in the model. For a quadratic antiferromagnet with easy-plane anisotropy the SGA is found to be the direct product SO(2,1)(X)SO(2,1) approximately SO(2,2). For the XY and the Heisenberg-Ising models of an antiferromagnetic linear chain the SGA is the compact counterpart of SO(2,1), i.e. SO(3) approximately SU(2). It is concluded that if the magnons are quantised as bosons then the SGA is the noncompact algebra SO(2,1), and if they are quantised as fermions then the SGA is the compact angular momentum algebra SO(3).
The exact renormalization-group approach of Wilson is used to study the critical behavior for
T>Tc,
H=0, and small $\ensuremath{\epsilon}>0$, of an isotropic ferromagnetic system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, with exchange and dipolar interactions between
d-component spins. Normal isotropic Heisenberg behavior with $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}\frac{\ensuremath{\epsilon}}{4}$ (to first order in $\ensuremath{\epsilon}$) is retained for $t=(\frac{T}{{T}_{c}})\ensuremath{-}1\ensuremath{\ll}\frac{G}{J{a}^{d}}$, where $G=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{2}$ measures the strength of the dipole-dipole interactions,
J is the short-range exchange parameter, and
a is the lattice spacing. When ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\frac{G}{J{a}^{d}}$, where $\ensuremath{\varphi}\ensuremath{\approx}1+\frac{\ensuremath{\epsilon}}{4}$, crossover occurs to a characteristic dipolar behavior described by a new fixed point of the recursion relations. For $t\ensuremath{\ll}\frac{G}{J{a}^{d}}$ one thus finds $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}\frac{9\ensuremath{\epsilon}}{34}$ {and, for spins of
nd components, $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}[\frac{(6n+3)}{2(6n+11)}]\ensuremath{\epsilon}$, which agrees with spherical-model results when $n\ensuremath{\rightarrow}\ensuremath{\infty}$}. In the dipolar regime the spin-correlation function $〈{s}_{q}^{\ensuremath{\alpha}}{s}_{\ensuremath{-}q}^{\ensuremath{\beta}}〉$ has a factor [${\ensuremath{\delta}}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{-}(\frac{{q}^{\ensuremath{\alpha}}{q}^{\ensuremath{\beta}}}{{q}^{2}})$], which suppresses longitudinal spin fluctuations; the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ display the expected demagnetization effects. It is found that dipolar anistropies derving from the lattice structure produce weak instabilities which should be hard to detect although their effects are not fully elucidated. Extensions of the results to nonzero magnetic fields, and to anistropic exchange interactions are indicated; the experimental situation is mentioned briefly.