Example. A three-by-three torus calculation [fdip-006S]
Example. A three-by-three torus calculation [fdip-006S]
For \(N_1=N_2=3\), there are 9 vertices, 18 edges and 9 plaquettes. The plaquette constraint permits only an even number \(k\) of negative plaquettes. There are \(\binom 9k\) such patterns and \(2^{10}=1024\) edge configurations for each pattern. The resulting numbers of edge configurations are \[\begin {array}{c|rrrrr} k&0&2&4&6&8\\\hline \text {count}&1024&36864&129024&86016&9216. \end {array}\] They sum to \(262144=2^{18}\), the number of all edge assignments.
Since \(\sum _pU_p=9-2k\), the partition polynomial is \[\begin {aligned} \mathcal Z(\beta )=1024\bigl (&e^{9\beta }+36e^{5\beta }+126e^\beta \\ &+84e^{-3\beta }+9e^{-7\beta }\bigr ),\\ Z_W(\beta )&=e^{-9\beta }\mathcal Z(\beta ). \end {aligned}\] At \(\beta =0\), this again gives \(2^{18}\). The factor \(e^{-9\beta }\) is necessary for the Wilson-action partition function even though it cancels from every normalized observable.
For \(\tanh \beta =1/2\), the exact loop formula gives \[\begin {aligned} \langle W(C_1)\rangle _W &=\frac {(1/2)+(1/2)^8}{1+(1/2)^9}=\frac {86}{171},\\ \langle W(C_2)\rangle _W &=\frac {(1/2)^2+(1/2)^7}{1+(1/2)^9}=\frac {44}{171}. \end {aligned}\] Here \(C_1\) bounds one square and \(C_2\) bounds two adjacent squares, each a simple disk boundary on this torus. These values include the finite-volume corrections; they are not yet \(t\) and \(t^2\).