Advanced Quantum Mechanics Schwabl · Problem 3.2

Correlation Functions, Scattering, and Response

Physics — Quantum Mechanics · heisenberg-ferromagnet-holstein-primakoff

Problem

The Heisenberg model of a ferromagnet is defined by the Hamiltonian H=−12∑l,l′J(∣l−l′∣)Sl⋅Sl′,H = -\frac{1}{2}\sum_{\mathbf{l},\mathbf{l}'}J(|\mathbf{l}-\mathbf{l}'|)\mathbf{S}_\mathbf{l}\cdot\mathbf{S}_{\mathbf{l}'}, where l\mathbf{l} and l′\mathbf{l}' are nearest neighbor sites on a cubic lattice. By means of the Holstein–Primakoff transformation Si+=2S φ(n^i)ai,S_i^+ = \sqrt{2S}\,\varphi(\hat{n}_i)a_i, Si−=2S ai†φ(n^i),S_i^- = \sqrt{2S}\,a_i^\dagger\varphi(\hat{n}_i), Siz=S−n^i,S_i^z = S - \hat{n}_i, with Si±=Six±SiyS_i^\pm = S_i^x \pm S_i^y, φ(n^i)=1−n^i/2S\varphi(\hat{n}_i) = \sqrt{1-\hat{n}_i/2S}, n^i=ai†ai\hat{n}_i = a_i^\dagger a_i and [ai,aj†]=δij[a_i, a_j^\dagger] = \delta_{ij}, [ai,aj]=0[a_i, a_j] = 0 one can express the Hamiltonian in terms of Bose operators aia_i. a) Show that the commutation relations for the spin operators are satisfied. b) Write down the Hamiltonian to second order (harmonic approximation) in terms of the Bose operators aia_i by regarding the square-roots in the above transformation as a shorthand for the series expansion. c) Diagonalize HH (by means of a Fourier transformation) and determine the dispersion relation of the spin waves (magnons).

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