Internal energy density of the critical three-state Potts model on the kagome lattice

It has been conjectured that the internal energy density of the Potts model on a semi-infinite strip with a width $L$ does not have any finite-size corrections at the critical point $K=K_c$. By factorizing the transfer matrix for the kagome lattice with larger widths, we have found that this conjecture is not correct in that the internal energy density slightly varies with $L$ at the critical point. From this size dependence of the internal energy density, we obtain an upper bound as $K_c < 1.0565615$, which is close to a recent estimate $K_c^{\rm JS} = 1.0565600(7)$ by Jacobsen and Scullard [arXiv:1204.0622]. We also obtain a lower bound as $K_c > 1.0560$ by calculating the correlation length along the strips.