Spin Waves in Ferromagnets

Consider the quantum mechanical spin-1/2 system with Hamiltonian

(i)

where the summation is over nearest-neighbor pairs in three dimensions.

a) Derive the equation of motion for the spin s_i at site of the lattice.

b) Convert the model to a classical microscopic model by inserting the

classical spin field s(r, t) into the equation of motion. Express to lowest order in its gradients, considering a simple cubic lattice with lattice constant a.

c) Consider the ferromagnetic case with uniform magnetization M = Mẑ. Derive the frequency-versus-wave vector relation of a small spin wave fluctuation s(r, t) = M + m₀(t) sin (k . r).

d) Quantize the spin waves in terms of magnons which are bosons. Derive the temperature dependence of the heat capacity.

SOLUTION

Quantum spins have the commutation relations

(1)

(2)

(3)

a) The time dependences of the spins are given by the equations of motion:

(4)

(5)

(6)

b) The classical spin field at point r_i is s(r_i, t). In the simple cubic lattice the six neighboring lattice sites are at the points r_j = r_i ± aĵ, where ĵ is  or ẑ. We expand the sum in a Taylor series, assuming that is a small number, and find

(7)

(8)

c) Given the form of the spin operator in part (c), one immediately derives the equation by neglecting terms of order O(m²₀):

(9)

(10)

(11)

(12)

The equations of motion have an eigenvalue ω_k, which represents the frequencies of the spin waves.

d) The internal energy per unit volume of the spin waves is given by

(13)

where the occupation number is suitable for bosons. At low temperature we can evaluate this expression by defining the dimensionless variable s = βhω_k, which gives for the integral

(14)

At low temperature the upper limit of the integral s₀ becomes large, and the internal energy is proportional to τ^5/2. The heat capacity is the derivative of with respect to temperature, so it goes as C ~ τ^3/2.