# Why does a dynamical gauge field accompany fractionalisation?

I'm trying to understand fractionalisation, of which spin-charge separation is an example. I've read that this is accomplished by introducing a Lagrange multiplier field, which becomes a dynamical gauge field at low energies.

My question is - does the gauge field emerge because we represent the spin degrees of freedom in terms of slave fermions, which are gauge-invariant, forcing the gauge field upon us? How is this related to the Lagrange multiplier field?

I'm also wondering what forces the gauge field to become dynamical at low energies?

I'd appreciate a good reference where this (or fractionalization and emergent gauge fields) is reviewed.

Essentially, yes, gauge fields emerge when you fractionalize your elementary particles. Suppose you have some sort of a constraint on your system e.g., no double occupancy at each site. $$\sum_{\sigma} a^\dagger_{i,\sigma} a_{i,\sigma} \leq 1$$ You can now write the electron operator in terms of spin and charge degrees of freedom $$c^\dagger_{i,\sigma} = b_{i} f_{i,\sigma}^\dagger$$ where $f_{i,\sigma}$ is a spinon and $b_{i}$ a holon. In terms of these, the constraint now becomes an equality, which is easier to deal with $$b_i^{\dagger}b_i + \sum_{\sigma}f^\dagger_{i,\sigma}f_{i,\sigma} = 1$$ In casting the electron as a composite, you've introduced a redundancy into the system since the electron operator is invariant local $U(1)$ transformations, $f \to e^{i \phi}f$ and $b\to e^{i\phi} b$. Thus, the gauge field is necessitated by fractionalization.