Field theory of collinear and noncollinear magnetic order

I thank the referee for a thorough reading of the manuscript and thoughtful comments. The lecture notes have been expanded to provide better explanations. Specific queries of the referee are addressed below.

1. Indeed, the antiferromagnetic chain has a topological contribution to the total spin that is not included when the uniform magnetization is integrated over the chain length. I have added a new subsection 4.5 Net spin of the antiferromagnetic chain to explain this effect and cited the seminal theoretical works of Faddeev and Takhtajan (for S=1/2) and of Papanicolaou (for classical spins).

2. The passage has been rewritten to provide a more explicit reference to the included term.

3. This seems a bit controversial! I have changed the offending terminology and now refer to the "analogs" of the Landau-Lifshitz equation for antiferromagnets or to simply "equations of motion."

4. An infinitesimal increment of a unit-vector field is orthogonal to the original value (so that the unit length is preserved), hence zero z component. In Sec. 3.2.1, I initially referred to a finite deviation; the subsequent Taylor expansion has no first-order term for the z-component.

5. Two sentences have been added to explain the meaning of topological sectors.

6. The term "elipsis" has been replaced with "omitted terms."

7. The text has been edited to explicitly state that Eq. (80) for the magnetization fields is the same as Eqs. (77) and (78) for three spins. (These are the updated equations numbers reflecting the addition of a new subsection.)