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We study the classical two-dimensional $\mathrm{RP^2}$ and Heisenberg models,using the Tensor-Network Renormalization (TNR) method. The determination of thephase diagram of these models has been challenging and controversial, owing tothe very large correlation lengths at low temperatures. The finite-sizespectrum of the transfer matrix obtained by TNR is useful in identifying theconformal field theory describing a possible critical point. Our resultsindicate that the ultraviolet fixed point for the Heisenberg model and theferromagnetic $\mathrm{RP^2}$ model in the zero temperature limit correspondsto a conformal field theory with central charge $c=2$, in agreement with twoindependent would-be Nambu-Goldstone modes. On the other hand, the ultravioletfixed point in the zero temperature limit for the antiferromagneticLebwohl-Lasher model, which is a variant of the $\mathrm{RP^2}$ model, seems tohave a larger central charge. This is consistent with $c=4$ expected from theeffective SO(5) symmetry. At $T >0$, the convergence of the spectrum is notgood in both the Heisenberg and ferromagnetic $\mathrm{RP^2}$ models. Moreover,there seems no appropriate candidate of conformal field theory matching thespectrum, which shows the effective central charge $c \sim 1.9$. These suggestthat both models have a single disordered phase at finite temperatures,although the ferromagnetic $\mathrm{RP^2}$ model exhibits a strong crossover atthe temperature where the dissociation of $\mathbb{Z}_2$ vortices has beenreported.

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