Compact U(1) Gauge Fields, Emergent Photons, and Monopoles

Compact U(1) gauge theory differs from noncompact Maxwell theory because the link angle is periodic and magnetic flux can change by monopole events. Whether an emergent photon survives depends decisively on spatial dimension, gauge-charged matter, and the scaling dimensions and symmetry quantum numbers of allowed monopoles.

Required background. Parton constraints supplies the emergent gauge charge; Coulomb, Higgs, and confining regimes supplies gauge-phase language.

Helpful background. Compact U(1) monopole plasma supplies the controlled pure-gauge confinement mechanism in $2+1$ dimensions.

Compact lattice electrodynamics

On an oriented link, let $a_\ell\sim a_\ell+2\pi$ and integer $E_\ell$ obey $[a_\ell,E_{\ell'}]=i\delta_{\ell\ell'}$. A representative Hamiltonian is

$$H=\frac{U}{2}\sum_\ell E_\ell^2 -K\sum_p\cos(\operatorname{curl}a)_p, \qquad (\nabla\cdot E)_i=\rho_i-\rho_i^{\rm bg}.$$

Expanding the cosine at weak fluctuations gives Maxwell theory and a transverse photon. Compactness is invisible in that expansion but permits configurations changing flux by $2\pi$. Their fugacity is a separate coupling.

Dimension changes the phase structure

In $3+1$ spacetime dimensions, weak compact coupling supports a Coulomb phase with $\omega=c|\mathbf k|$, power-law field correlations, and gapped electric charges and magnetic monopoles. Increasing fluctuations can proliferate monopoles and confine.

In $2+1$ dimensions, monopoles are instanton events. In pure compact U(1), their dilute gas screens the dual photon and produces confinement at all couplings Polyakov 1977. Therefore a $2+1$D parton mean-field photon is not stable merely because the quadratic action is Maxwell-like.

Gapless matter can increase a monopole operator’s scaling dimension. At a putative $2+1$D fixed point, an allowed monopole of dimension $\Delta_{\mathcal M}$ is irrelevant only if $\Delta_{\mathcal M}>3$. Lattice symmetry may forbid the lowest charge and allow only doubled or quadrupled monopoles. Large-flavor calculations can be controlled, but extrapolating to physical flavor number requires independent checks.

Observables and caveats

An emergent photon is detected through gauge-invariant field correlations, thermodynamics, and characteristic pinch-point structure—not through $\langle a_\ell\rangle$. Monopole quantum numbers determine which conventional order appears after confinement. Dynamical charge can screen Wilson lines, so one combines loop response with charge/monopole gaps and topological sectors.

At nonzero temperature, a quantum Coulomb regime may cross over rather than define a sharp phase, depending on dimension and defects. Disorder can broaden photon response or nucleate charges. Every claim should state the photon bandwidth, charge gap, monopole scale, and temperature window.

Exercise

A monopole operator at a proposed $2+1$D fixed point has $\Delta_{\mathcal M}=3.4$. Is its fugacity relevant at linear order?

Solution

Its coupling has RG eigenvalue $y=3-\Delta_{\mathcal M}=-0.4$, so it is linearly irrelevant. This conclusion assumes that this is the lowest symmetry-allowed monopole and that the fixed point itself exists and is stable to all other perturbations.

References

• Alexander M. Polyakov, “Quark Confinement and Topology of Gauge Theories,” Nuclear Physics B 120 (1977) 429–458, doi:10.1016/0550-3213(77)90086-4.