Computational approach to scaling and criticality in planar Ising models

Abstract

In this thesis, we study the critical behaviour of the two-dimensional Ising model on the regular lattices. Using the numerical solution of the model on the square, triangular and honeycomb lattices we compute the universal scaling function, which turns out to be identical on each of the lattices, in addition to being identical to the scaling function of the Ising Field Theory, computed previously by Fonseca and Zamolodchikov. To cope with the lattice contributions we carefully examined series expansions of the lattice free energy derivatives. We included the non-scaling regular part of the free energy as well as non-linear Aharony-Fisher scaling elds, which all have non-universal expansions. Using as many of the previously known exact results as possible, we were able to t the unknown coe cients of the scaling function expansion and obtain some non-universal coe cients. In contrast to the IFT approach of Fonseca and Zamolodchikov, all coe cients were obtained independently from separate datasets, without using dispersion relations. These results show that the Scaling and Universality hypotheses, with the help of the Aharony-Fisher corrections, hold on the lattice to very high precision and so there should be no doubt of their validity. For all numerical computations we used the Corner Transfer Matrix Renormalisation Group (CTMRG) algorithm, introduced by Nishino and Okunishi. The algorithm combines Baxter's variational approach (which gives Corner Transfer Matrix (CTM) equations), and White's Density Matrix Renormalisation Group (DMRG) method to solve the CTM equations e ciently. It was shown that given su cient distance from the critical point, the algorithmic precision is exceptionally good and is unlikely to be exceeded with any other general algorithm using the same amount of numerical computations. While performing tests we also con rmed several critical parameters of the three-state Ising and Blume-Capel models, although no extra precision was gained, compared to previous results from other methods. In addition to the results presented here, we produced an efficient and reusable implementation of the CTMRG algorithm, which after minor modifications could be used for a variety of lattice models, such as the Kashiwara-Miwa and the chiral Potts models.