Two-plaquette SU(2) rectangle

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Take a free-boundary $SU(2)$ square lattice containing a rectangle two plaquettes wide and one plaquette high, and let $C$ follow its oriented face boundary counterclockwise from the lower-left corner. Its loop holonomy $U_C$ is the ordered link product, using inverse matrices on reverse edges, and the defining normalized Wilson loop is $W(C):=\frac{1}{2}{\mathrm{Tr}}_2(U_C)$. For $0<\beta \le 1$, the enclosed area is $A=2$ plaquettes, so the exact finite-lattice expectation is $⟨W(C){⟩}_{\beta }=\lambda (\beta {)}^2={\beta }^2\mathrm{/}16+O({\beta }^4)$ as $\beta \downarrow 0$, where

is the finite one-plaquette coefficient for ${\mu }_2$, the normalized pushforward of round $S^3$ surface area under the matrix-coordinate map in the finite $SU(2)$ plaquette-measure definition.

Exact finite-lattice Wilson area formula

One-plaquette strong-coupling coefficient

Meyer, §1.4.1.1, printed pp. 22–24, on the leading minimal plaquette tiling

Lohmayer, §4.2, printed pp. 61–63, on exact rectangular Wilson-loop area dependence

The black arrows are positively oriented links (east along rows and north along columns). The red arrows trace the counterclockwise boundary $C$; the top and left portions therefore use inverse link variables. The labels $p_{0,0}$ and $p_{1,0}$ are the two plaquettes, and the area count is two. The proof below uses only the ordered products and the theorem, not the diagram.