  In this paper, we show that spin waves, the elementary excitation of the
Heisenberg magnetic system, obey a kind of intermediate statistics with a
finite maximum occupation number n. We construct an operator realization for
the intermediate statistics obeyed by magnons, the quantized spin waves, and
then construct a corresponding intermediate-statistics realization for the
angular momentum algebra in terms of the creation and annihilation operators of
the magnons. In other words, instead of the Holstein-Primakoff representation,
a bosonic representation subject to a constraint on the occupation number, we
present an intermediate-statistics representation with no constraints. In this
realization, the maximum occupation number is naturally embodied in the
commutation relation of creation and annihilation operators, while the
Holstein-Primakoff representation is a bosonic operator relation with an
additional putting-in-by-hand restriction on the occupation number. We deduce
the intermediate-statistics distribution function for magnons. On the basis of
these results, we calculate the dispersion relations for ferromagnetic and
antiferromagnetic spin waves. The relations between the intermediate statistics
that magnons obey and the other two important kinds of intermediate statistics,
Haldane-Wu statistics and the fractional statistics of anyons, are discussed.
We also compare the spectrum of the intermediate-statistics spin wave with the
exact solution of the one-dimensional s = 1/2 Heisenberg model, which is
obtained by the Bethe ansatz method. For ferromagnets, we take the
contributions from the interaction between magnons (the quartic contribution),
the next-to-nearest neighbor interaction, and the dipolar interaction into
account for comparison with the experiment.
