Magnetism: Heisenberg Model & Spin Waves
Spin-spin interactions, ferro/antiferromagnetism, and magnon excitations.
The Heisenberg Hamiltonian on a lattice:
$$H = -J \sum_{\langle ij\rangle} \mathbf S_i \cdot \mathbf S_j - h \sum_i S_i^z.$$$J > 0$ favors parallel alignment (ferromagnet); $J < 0$ favors antiparallel (antiferromagnet). Quantum spins $\mathbf S_i$ satisfy $[S_i^a, S_j^b] = i\hbar \delta_{ij} \epsilon^{abc} S_i^c$.
Ground state of a 1D ferromagnetic chain is the fully-polarized $|\uparrow\uparrow\cdots\uparrow\rangle$. Single spin-flip excitations are magnons — collective spin waves with dispersion
$$\hbar\omega(k) = 4 J S (1 - \cos(ka)).$$Spin waves are bosonic Goldstone modes of the spontaneously broken $SU(2)$ rotational symmetry.
1D Heisenberg antiferromagnet: ground state is a complicated singlet (Bethe ansatz, 1931). Mermin–Wagner: continuous symmetry breaking is impossible in $d \leq 2$ at finite $T$ — no long-range order in 2D Heisenberg ferromagnets, but Ising (discrete) ones order fine.