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

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 ?

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.

• Thanks a lot. So in the $CP^1$ representation, the emergence of spinons can be seen very neatly. Is there any way we could have anticipated from the $\vec{n}$ representation, that there can be these spin 1/2 particles ? Mar 9 '16 at 13:00
• @TuhinSubhraMukherjee Yes, that is the resonance valence bond (RVB) picture of the spin liquid. The RVB state is an ideal ground state wave function of spin liquid, which is a superposition of all spin-singlet dimmers. To make an excitation, one can break a dimmer which creates two dangling spins, which are two spinons, each carrying spin-1/2. Due to the RVB background, the spinons are deconfined, as they can be separated far apart by dimmer resonance. Then locally you obtain a single spinon. Of corse globally the spinon must come in pairs, this is why the spin liquid order is topological. Mar 10 '16 at 19:27
• @TuhinSubhraMukherjee By the way, the gauge boson (emergent electromagnetic wave) also have an interpretation in the RVB picture: they are collective fluctuations of the dimmer patterns. It turns out that each dimmer covering on a lattice corresponds to a closed string configuration on the same lattice, via a mapping called transition graph. The strings are then interpreted as electric field lines, and the spinons are end points of the string, which naturally carries the gauge charge and couples to the gauge field. Mar 10 '16 at 19:35
• @TuhinSubhraMukherjee No, RVB state is not a condensate of spinons. RVB state is a deconfined state, where both spinons and visons are not condensed. Spinon condensation will lead to anti-ferromagnetic (AFM) state, and vison condensation will lead to the valence bond solid (VBS) state. I believe what people talked about is the VBS state as vison condensation. Apr 27 '16 at 7:12
• @TuhinSubhraMukherjee They actually mean RVB state = condensate of spinon Cooper pairs, but not the condensate of spinons themselves. BCS condensate of some quasiparticles actually means to condense the Cooper pairs of those quasiparticles. By such a BCS condensation, the U(1) gauge structure is Higgs down to $Z_2$, which leads to $Z_2$ topological ordered RVB spin liquid. But in the $Z_2$ spin liquid, spinons are still deconfined and not condensed. Apr 28 '16 at 4:28