One-dimensional three-spin cell calculation for two-dimensional Ising exponents

We have calculated the three-spin cell cumulant expansion of Niemeijer and Van Leeuwen for the one-dimensional $\mathrm{XY}$ model in order to obtain the critical exponents of the two-dimensional Ising model. We found that the calculation of this type is not as accurate as the direct cumulant-expansion calculation for the two-dimensional Ising model. Specifically, the results of our present calculation deviate considerably from the exact results in comparison to the results obtained by Hsu, Niemeijer, and Gunton. Moreover, we find that the odd-spin eigenvalue is not physical in this particular renormalization-group transformation.