 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-81/1/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

Repeat the linear spin-wave approximation for bonds of lengths one and two. The polarized energy is

E_{0} = - N S^{2} (J_{1} + J_{2})

and each bond length contributes its own lattice Laplacian. Thus

ε_{k} = 2 S [J_{1} (1 - cos k) + J_{2} (1 - cos 2 k)] = 4 S [J_{1} sin^{2} (k /2) + J_{2} sin^{2} k] .

The next-nearest-neighbour ferromagnetic magnon dispersion remains nonnegative for the stated positive exchanges and has its only zero at

k = 0

2 π

when

J_{1} > 0

. At small

k

ε_{k} ≃ S (J_{1} + 4 J_{2}) k^{2}

: longer-range ferromagnetic exchange increases the stiffness.

At this harmonic order, a bosonic occupation configuration has energy

E_{0} + \sum_{k} ε_{k} n_{k}

. The one-magnon dispersion is also exact. General multi-magnon energies acquire the interactions omitted in the large-

S

expansion; the formula does not claim an exact diagonalization of the full interacting many-spin Hamiltonian.

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