Why does the Heisenberg Hamiltonian have a minus sign?

[matematikuvol]

Why is minus sine in definition of hamiltonian

$$H=-\sum_{i,j}J_{i,j}(S_{i}^+S_{j}^-+S_i^zS_j^z)$$

Why not?

$$H=\sum_{i,j}J_{i,j}(S_{i}^+S_{j}^-+S_i^zS_j^z)$$

 

[Physics Monkey]

The choice of sign is determined by the physics. If the spins want to align (ferromagnetic) then you should have H = |J| S*S while if the spins want to anti-align (antiferromagnetic) then H=-|J|S*S is correct. Of course, you can always write H=-J S*S and then set J>0 for ferro and J<0 for anti-ferro. That's just a convention.