Parton Constructions and Gauge Constraints

A parton construction factorizes a physical operator into simpler fields at the price of enlarging the Hilbert space. The redundancy of that factorization becomes an emergent gauge structure, and a local constraint—equivalently Gauss law—projects back to physical states. Mean-field bands are meaningful only together with this constraint and a stability analysis of gauge fluctuations.

Required background. Gauge redundancy and observables distinguishes descriptions from observables; gauge orbits and Gauss constraints supplies the physical-state condition; the Hubbard Mott regime supplies localized spins from constrained electrons.

Helpful background. Gauge-invariant dressed observables supplies physical composite operators; emergent variables supplies effective-description logic.

Abrikosov fermions and the local constraint

For spin $1/2$,

$$S_i^a=\frac12 f_{i\alpha}^\dagger\sigma^a_{\alpha\beta}f_{i\beta}, \qquad n_{fi}=1.$$

The parton Fock space has empty and doubly occupied states absent from the physical spin Hilbert space. The generator $G_i=n_{fi}-1$ enforces $G_i|\mathrm{phys}\rangle=0$, and $f_{i\alpha}\mapsto e^{i\theta_i}f_{i\alpha}$ leaves every $S_i^a$ unchanged. The U(1) phase is a redundancy, not a microscopic global symmetry.

Decoupling exchange introduces hopping $\chi_{ij}=\langle f_i^\dagger f_j\rangle$ and pairing $\Delta_{ij}=\langle f_i\epsilon f_j\rangle$. A gauge transformation changes these link fields. The subgroup that leaves an ansatz invariant—the invariant gauge group—can be SU(2), U(1), or Z2. Fluctuations about the ansatz supply the corresponding gauge field Wen 2002.

Projection and physical observables

A variational state is

$$|\Psi_{\rm phys}\rangle=\mathcal P_G|\Psi_{\rm MF}\rangle, \qquad \mathcal P_G=\prod_i\delta_{n_{fi},1}.$$

Projection can qualitatively change correlations and spectra. A parton Green function is gauge dependent; neutron scattering probes spin bilinears, and a physical electron in a slave-particle construction is a gauge-neutral composite. Gauge fluctuations can bind partons, broaden continua, or confine them altogether.

In a slave-boson electron decomposition such as $c_{i\sigma}=b_i^\dagger f_{i\sigma}$, the constraint links boson and spinon number. Electromagnetic and emergent gauge charges must be tabulated separately. Assigning the electron conductivity directly to $f$ violates the constraint; response composition includes the emergent gauge field.

Validity tests

State the exact operator identity, enlarged states, constraint, gauge group, compactness, and symmetry action. Check whether the projected state is nonzero, whether the claimed phase survives gauge fluctuations and allowed monopoles, and whether gauge-invariant correlations match the proposal. Different ansätze related by gauge transformations are one physical state; distinct-looking parton dispersions need not represent distinct phases.

Exercise

Show explicitly that $\mathbf S_i$ is invariant under $f_{i\alpha}\mapsto e^{i\theta_i}f_{i\alpha}$.

Solution

$f_{i\alpha}^\dagger\mapsto e^{-i\theta_i}f_{i\alpha}^\dagger$ and $f_{i\beta}\mapsto e^{i\theta_i}f_{i\beta}$, so the phases cancel in $f_i^\dagger\sigma^a f_i$. The parton itself is gauge charged; the spin bilinear is physical.

References

• Xiao-Gang Wen, “Quantum Orders and Symmetric Spin Liquids,” Physical Review B 65 (2002) 165113, doi:10.1103/PhysRevB.65.165113.