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Tensor Network Renormalization Study on the Crossover in Classical Heisenberg and RP² Models in Two Dimensions
Atsushi Ueda, Masaki Oshikawa
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We study the classical two-dimensional RP² 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 RP² 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 RP² 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 RP² models. Moreover, there seems 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 RP² model exhibits a strong crossover at the temperature where the dissociation of ℤ₂ vortices has been reported.
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