Antiferromagnetism

Antiferromagnetism

For a bipartite lattice with $J_{ij}\geq 0$, one can assume a Néel state that is the fully polarized ground state with the opposite directions for the two sublattices. It is written as the product of the eigenstates at two different sublattices:

\[\begin{equation} \ket{\Psi_{\rm GND}} = \prod_{i\in A}\ket{S}_{i}\prod_{j\in B}\ket{-S}_{j} \end{equation}\]

where the state $\ket{S}$ is the eigenstate of $S^{z}$ for the given lattice site with eigenvalues of $\pm S$. Note that $\ket{S}_{i}$ and $\ket{-S}_{j}$ are the eigenstates with the maximum value of $S$. Thus, if one applies $S^{+}_{i}S^{-}_{j}$ to the Néel state, one gets: $S^{+}_{i}S^{-}_{j}\ket{\Psi_{\rm GND}} = 0$. Acting the Hamiltonian $H$ onto the ground state, one obtains the following relation:

\[\begin{align} H\ket{\Psi_{\rm GND}} &= \sum\limits_{\braket{ij}}J_{ij}\left[\frac{1}{2}(S^{+}_{i}S^{-}_{j} + S^{-}_{i}S^{+}_{j}) + S^{z}_{i}S^{z}_{j}\right]\prod_{i\in A}\ket{S}_{i}\prod_{j\in B}\ket{-S}_{j} \nonumber \\ &= \sum\limits_{\braket{ij}}J_{ij}\left[\frac{1}{2}S^{-}_{i}S^{+}_{j} - S^{2}\right]\prod_{i\in A}\ket{S}_{i}\prod_{j\in B}\ket{-S}_{j} \nonumber \\ &= \sum\limits_{\braket{ij}}J_{ij}\left[\frac{1}{2}\sqrt{S(S+1)-S(S-1)}\prod_{i\in A}\ket{S-1}_{i}\sqrt{S(S+1)-S(S-1)}\prod_{j\in B}\ket{-S+1}_{j} - S^{2}\prod_{i\in A}\ket{S}_{i}\prod_{j\in B}\ket{-S}_{j}\right] \nonumber \\ &= \sum\limits_{\braket{ij}}J_{ij}\left[\frac{1}{2}{S(S+1)-S(S-1)}\prod_{i\in A}\ket{S-1}_{i}\prod_{j\in B}\ket{-S+1}_{j} - S^{2}\prod_{i\in A}\ket{S}_{i}\prod_{j\in B}\ket{-S}_{j}\right] \end{align}\]

Here one can notice that $\ket{\Psi_{\rm GND}}$ is not the eigenstate of the Hamiltonian. The true eigenstate of the Hamiltonian satisfies the so-called Marshall’s theorem, which states that the ground state for equal-size sublattices is a singlet of the total spin ($S_{\rm tot}\ket{\Psi_{\rm GND}} = 0$). Note the number of singlets grows exponentially as a function of the lattice size.