Lieb–Schultz–Mattis theorem

Remarks

The idea of the proof is a twist. Let $\psi$ be the ground state, which for the Heisenberg antiferromagnet on an even ring is a unique total-spin singlet invariant under translation up to a phase. Set $U = \exp\big(\tfrac{2\pi i}{L}\sum_{j} j\, S^z_j\big)$, which rotates the spins slowly along the chain. Each bond energy changes by $O(1/L^2)$, so $\langle U\psi, H_L U\psi \rangle - E_0(L) = O(1/L)$. The translation $T$ satisfies $TUT^{-1} = U e^{-2\pi i S^z_{\mathrm{tot}}/L} e^{2\pi i S^z_1}$, and $e^{2\pi i S^z_1} = -1$ for half-odd-integer $S$; since $S^z_{\mathrm{tot}}\psi = 0$ and $T\psi$ is a multiple of $\psi$, this gives $\langle \psi, U\psi \rangle = -\langle \psi, U\psi \rangle$, so $\varphi = U\psi$ is orthogonal to $\psi$. Lieb, Schultz and Mattis proved the case $S = \tfrac12$ in 1961, and Affleck and Lieb extended it to all half-odd-integer $S$ in 1986. The argument fails for integer $S$, where $e^{2\pi i S^z_1} = 1$, and indeed the spin-$1$ AKLT chain has a gap.

Going deeper

For a student

The result is a constraint from symmetry and filling: it relates translation symmetry, spin rotation symmetry and the half-integer spin per unit cell. For the spin-$\tfrac12$ chain the exact solution confirms the gapless option. For integer spins the argument breaks down, and Haldane's prediction of a gap is consistent with it. Modern generalizations, by Oshikawa and by Hastings in higher dimensions, give similar constraints for lattices with a fractional filling per unit cell.