Tensor Network Renormalization Study on the Crossover in Classical Heisenberg and $\mathrm{RP^2}$ Models in Two Dimensions

We study the classical two-dimensional $\mathrm{RP^2}$ and Heisenberg models, using the Tensor-Network Renormalization (TNR) method. The determination of the phase diagram of these models has been challenging and controversial, owing 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 $\mathrm{RP^2}$ 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 $\mathrm{RP^2}$ 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 $\mathrm{RP^2}$ models. Moreover, there seems no appropriate candidate of conformal field theory matching the spectrum, which shows the effective central charge $c \sim 1.9$. These suggest that both models have a single disordered phase at finite temperatures, although the ferromagnetic $\mathrm{RP^2}$ model exhibits a strong crossover at the temperature where the dissociation of $\mathbb{Z}_2$ vortices has been reported.