Theorems in magnetism

Theorems in magnetism

This note shows the summary of well-known theorems in magnetism. The proofs for these theorems are not discussed yet.

Goldstone’s theorem. With the locality condition $|J_{ij}| |\mathbf{r}_{i}-\mathbf{r}_{j}| < \infty$, if a spin structure factor

\[\begin{equation} \sum\limits_{\braket{ij}}e^{-i\mathbf{k}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}\braket{\mathbf{S}_{i}\cdot\mathbf{S}_{j}} \end{equation}\]

diverges at $\mathbf{k}=\mathbf{Q}$ due to continuous symmetry breaking, there exists an eigenstate with the momentum $\mathbf{k}$, where the eigenvalue $E(\mathbf{k})$ vanishes at $\mathbf{Q}$. Note that the presence of a gapless excitation does not necessarily mean that a continuous symmetry is broken.

Mermin-Wagner theorem.

In one-dimensional Heisenberg model with the locality condition, there must be no spontaneously symmetry breaking. In two-dimension, there must be no spontaneously symmetry breaking at finite temperature.

\[\begin{equation} \lim_{h\rightarrow 0}\lim_{N\rightarrow \infty}\braket{m} = 0 \end{equation}\]

Nagaoka theorem.

For the system with $U\rightarrow\infty$ and with one extra hole or electron in a bipartite lattice, it favors ferromagnetism. This theorem assumes a situation where the number of electrons is fixed and the lattice size is finite.

Marshall’s theorem.

For the antiferromagnetic Heisenberg model on the bipartite lattice, the total ground state should be a singlet state ($\braket{S} = 0$).

Lieb-Schultz-Mattis theorem.

For the $S=1/2$ antiferromagnetic Heisenberg model on the bipartite one-dimensional lattice (satisfying periodic boundary condition), the ground state should be a singlet state with a gapless spectrum or a doubly degenerate state with a gapped spectrum. This implies that there is no unique ground state for a gapped system in thermodynamic limit, but degenerate state below the gap.