Equivalence of nonlinear sigma model and the $CP^1$ model

Question

While studying the non-linear sigma model, defined by the action $\mathcal{S} = \int dtd^2x (\partial_\mu n^a \partial^\mu n^a)$ along with the constraint $n^a n^a=1$, people often use the map $n^a = z^\dagger \sigma^a z$, where $z$ is a two-component spinor satisfying $z^\dagger z=1$. So the space of $\vec{n}$ is a two-sphere ($S^2$), while space of $z$ is a three-sphere ($S^3$). I do not understand what one achieves by going from $\vec{n}$ to $z$. Is there some properties of the theory that are more easy to see in terms of the $z$ fields ? In other words why $S^3$ is "better" than $S^2$ for studying the theory ?

Answer 1

More precisely speaking, the theory of the O(3) vector $\vec{n}$ is equivalent to the theory of spinor $z$ with the U(1) gauge field $a$. They are just two descriptions of the same physical system, and no one is better than the other.

Note that the gauge field in the $CP^1$ model is crucial. Because the space of the spinor $S^3$ (unphysical) is larger than that of the spin $S^2$ (physical), implying that the spinor description contains some redundancy, the redundancy must be "gauged away" by the U(1) gauge field. This is the standard formalism of fractionalization: one rewrites the physical degree of freedom (the spin $\vec{n}$) as fractionalized degree of freedom (the spinon $z$) with gauge structures. If the gauge theory is in its confined phase, then the spinons are combined into the spin, and we gain nothing from fractionalization.

However, if the gauge theory is in a deconfined phase, we can have gauge bosons and deconfined spinons as the low-energy emergent excitations of the spin system, which is not easy to see from the non-linear sigma model. In this case, the spin system is said to be in a spin-liquid phase with the topological order. It is further possible to condense spinon-pairs, and Higgs the U(1) gauge structure to $Z_2$, which is a different kind of topological order. So what we gain from rewriting the non-linear sigma model as the $CP^1$ theory is the description power of various topological orders in the exotic spin-liquid states.