Tensor network renormalization study on the crossover in classical Heisenberg and RP^{2} models in two dimensions

We study the classical two-dimensional RP^{2} and Heisenberg models, using the tensor-network renormalization (TNR) method. The determination of the phase diagram of these models has been challenging and controversial due to the very large correlation lengths at low temperatures. The finite-size spectrum of the transfer matrix obtained by TNR is useful in identifying the conformal field theory describing a possible critical point. Our results indicate that the ultraviolet fixed point for the Heisenberg model and the ferromagnetic RP^{2} model in the zero-temperature limit corresponds to a conformal field theory with central charge c=2, in agreement with two independent would-be Nambu-Goldstone modes. On the other hand, the ultraviolet fixed point in the zero-temperature limit for the antiferromagnetic Lebwohl-Lasher model, which is a variant of the RP^{2} model, seems to have a larger central charge. This is consistent with c=4 expected from the effective SO(5) symmetry. At T>0, the convergence of the spectrum is not good in both the Heisenberg and ferromagnetic RP^{2} models. Moreover, there seems to be no appropriate candidate of conformal field theory matching the spectrum, which shows the effective central charge c∼1.9. These suggest that both models have a single disordered phase at finite temperatures, although the ferromagnetic RP^{2} model exhibits a strong crossover at the temperature where the dissociation of Z_{2} vortices has been reported.