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

Abstract

We study the classical two-dimensional $\mathrm{R}{\mathrm{P}}^2$ 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 $\mathrm{R}{\mathrm{P}}^2$ 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 $\mathrm{R}{\mathrm{P}}^2$ 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 $\mathrm{R}{\mathrm{P}}^2$ 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 $\mathrm{R}{\mathrm{P}}^2$ model exhibits a strong crossover at the temperature where the dissociation of ${\mathbb{Z}}_2$ vortices has been reported.

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• Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and ${\mathrm{RP}}^2$ models: zero-temperature phase transition versus finite-temperature crossover
• Asymptotic Freedom and Finite-size Scaling of Two-dimensional Classical Heisenberg Model
• On the $R{P}^{N-1}$ and $C{P}^{N-1}$ universality classes
• Characterizing spin ordering via maximal row correlation in classical spin models
• Nonpertubative Many-Body Theory for the Two-Dimensional Hubbard Model at Low Temperature: From Weak to Strong Coupling Regimes