In charge-n monopole theories on a twisted cylinder, even n gives complete screening with emergent ±2/n charges, while odd n keeps a reduced-tension confining string.
Interfaces, Strings, and a Soft Mode in the Square Lattice Quantum Dimer Model
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The quantum dimer model on the square lattice is equivalent to a $U(1)$ gauge theory. Quantum Monte Carlo calculations reveal that, for values of the Rokhsar-Kivelson (RK) coupling $\lambda < 1$, the theory exists in a confining columnar phase. The interfaces separating distinct columnar phases display plaquette order, which, however, is not realized as a bulk phase. Static "electric" charges are confined by flux tubes that consist of multiple strands, each carrying a fractionalized flux $\frac{1}{4}$. A soft pseudo-Goldstone mode emerges around $\lambda \approx 0$, long before one reaches the RK point at $\lambda = 1$.
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Fractionalization of flux tubes in 3d and screening by emergent electric charges in 2d
In charge-n monopole theories on a twisted cylinder, even n gives complete screening with emergent ±2/n charges, while odd n keeps a reduced-tension confining string.