Spectrum of the Anti-Ferromagnet and Ferromagnet:
While we will come to more details later, let us just quickly make some statements that seem to make sense. If you have a Heisenberg AF in 2-spatial dimensions, then the ground state (GS) is not exactly a Neel state, since there are quantum fluctuations which destabilize the Neel state. Just to have an idea of the numbers, note that a two-spin system can form a singlet and triplet. The singlet has lower energy, which is -3 J /4. Next consider a large system with the pure Neel state -- this has an energy density of -J/4. Now, it is clear that because the pure Neel state is not an eigenstate of AF Heisenberg model (AFHM), the GS energy must be lower; it is actually close to -0.670 J. This makes it clear that the GS of the AFHM is a superposition of spin-singlets formed by a spin with it's different neighbors, --- a single spin cannot form a singlet with more than a single neighbor.
The excitations above the GS would be the so-called "rotor states" and they would carry spin. I will add more about them later.
While we will come to more details later, let us just quickly make some statements that seem to make sense. If you have a Heisenberg AF in 2-spatial dimensions, then the ground state (GS) is not exactly a Neel state, since there are quantum fluctuations which destabilize the Neel state. Just to have an idea of the numbers, note that a two-spin system can form a singlet and triplet. The singlet has lower energy, which is -3 J /4. Next consider a large system with the pure Neel state -- this has an energy density of -J/4. Now, it is clear that because the pure Neel state is not an eigenstate of AF Heisenberg model (AFHM), the GS energy must be lower; it is actually close to -0.670 J. This makes it clear that the GS of the AFHM is a superposition of spin-singlets formed by a spin with it's different neighbors, --- a single spin cannot form a singlet with more than a single neighbor.
The excitations above the GS would be the so-called "rotor states" and they would carry spin. I will add more about them later.
