Let us consider the AFM (antiferromagnetism) Heisenberg Hamiltonian:

$$H=+J\sum_{<i,j>} \hat{S}_i \cdot \hat{S}_j$$

in any dimension, in the square lattice. Say 1 spatial dim on a closed chain, 2 spatial dim with square lattice, 3 spatial dim with cubic lattice, etc.

The spin $ \hat{S}_i$ can be integer or half-integer spin.

**question 1: Do we have the ground states Neel order in any dimensions for this Hamiltonian?**

**question 2: Do we have the gapless spin wave in any dimensions for this Hamiltonian? Regardless the spin** as integer or half-integer spin?

**question 3: Do the Haldane gap phenomenon in any dimensions for this Hamiltonian? **integer or half-integer spin??** (I know the energy gap occurs at the AKLT Hamiltonian for spin-1/2, but I doubt it is the case for this **Heisenberg Hamiltonian.**) **