Comment on “ Duality relations for Potts correlation functions ”

In a recent paper by Wu [1] the three-point correlation of the q-state Potts model on a planar graph was related to ratios of dual partition functions under fixed boundary conditions. It was claimed that the method employed could straightforwardly be applied to higher correlations as well; this is however not true. By explicitly considering the four-point correlation we demonstrate how the appearence of non-well-nested connectivities invalidates the method. Consider the q-state Potts model [2] on a two-dimensional planar graph L having a free boundary. In accordance with Fig. 2 in Ref. [1] we let i, j, k and l be four sites on the boundary of L, following one another in a clockwise fashion, and we define exterior dual spins s1, s2, s3 and s4 so that all boundary spins between sites l and i of L interact with a spin state s1, boundary spins between i and j interact with s2, spins between j and k with s3, and finally spins between k and l with s4. The partition function with the four Potts spins σi, σj , σk and σl fixed in definite states is called Zσiσjσkσl , and similarly the dual partition function for fixed exterior dual spins is denoted by Z s1s2s3s4. Up to the q-fold permutation symmetry of the Potts spin labels there exist 15 different boundary conditions for Zσiσjσkσl out of which we can form five combinations 1 Z4 = Z1111, Z3 = Z2111 + Z1211 + Z1121 + Z1112, Z2p = Z2211 + Z2121 + Z2112, Zp = Z1123 + Z1213 + Z1231 + Z2113 + Z2131 + Z2311, Z0 = Z1234, (1) which are symmetric under permutations of the four sites i, j, k and l. We introduce them here in order to simplify the notation in subsequent equations. Following Wu’s strategy [1] we should extract equations relating the Zσiσjσkσl and the Z s1s2s3s4 by running through all possible ways of connecting the sites i, j, k and l by auxiliary bonds and using the fundamental duality relation, Wu’s Eq. (3). Each time a bond is added one of the exterior dual spins s is separated from its neighbours and thus allowed to take a different value. First, using the ‘empty’ connection (i.e., introducing no auxiliary bonds) we find the relation Z4 + (q − 1)[Z3 + Z2p] + (q − 1)(q − 2)Zp + (q − 1)(q − 2)(q − 3)Z0 = qCZ ∗ 4 , (2) corresponding to Wu’s Eq. (19). Next, consider adding a bond between sites k and l. Summing over the two ‘free’ sites, i and j, we obtain a reduction to the two-point case: