Hi, I just started self studying solid state and I'm having trouble figuring out what the hamiltonian for a square lattice would be when considering the heisenberg interaction. I reformulated the dot product into 1/2( Si+Si+δ+ +Si+δ+S-- ) + SizSi+δz and use Siz = S-ai+ai Si+ = √2S]ai ... Si+δz=-S+ai+δ+ai+δ ... Etc. But I'm getting for the terms of the hamiltonian aiai+δ +ai+δ+ai+ .... but don't these terms violate momentum conservation? What is the real heisenberg interaction hamiltonian for the square lattice?