Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and $\mathrm{RP}^2$ models: zero-temperature phase transition versus finite-temperature crossover

TL;DR

This study uses tensor-network methods to compare the two-dimensional classical Heisenberg and RP^2 models, revealing a zero-temperature phase transition in the Heisenberg model and a finite-temperature crossover in the RP^2 model, highlighting fundamental differences in their critical behavior.

Contribution

The paper introduces a tensor-network approach with explicit SO(3) symmetry to analyze correlation lengths and entanglement spectra in both models, clarifying their distinct critical phenomena.

Findings

Heisenberg model shows no finite-temperature phase transition, supporting asymptotic freedom.

RP^2 model exhibits a crossover with strong nematic quasi-long-range order.

Scaling behavior differs: zero-temperature shift in Heisenberg, finite-temperature crossover in RP^2.

Abstract

Tensor-network methods are used to perform a comparative study of the two-dimensional classical Heisenberg and $\mathrm{RP}^2$ models. We demonstrate that uniform matrix product states (MPS) with explicit $\mathrm{SO}(3)$ symmetry can probe correlation lengths up to $\mathcal{O}(10^3)$ sites accurately, and we study the scaling of entanglement entropy and universal features of MPS entanglement spectra. For the Heisenberg model, we find no signs of a finite-temperature phase transition, supporting the scenario of asymptotic freedom. For the $\mathrm{RP}^2$ model we observe an abrupt onset of scaling behaviour, consistent with hints of a finite-temperature phase transition reported in previous studies. A careful analysis of the softening of the correlation length divergence, the scaling of the entanglement entropy and the MPS entanglement spectra shows that our results are inconsistent with true criticality, but are rather in agreement with the scenario of a crossover to a pseudo-critical region which exhibits strong signatures of nematic quasi-long-range order at length scales below the true correlation length. Our results reveal a fundamental difference in scaling behaviour between the Heisenberg and $\mathrm{RP}^2$ models: Whereas the emergence of scaling in the former shifts to zero temperature if the bond dimension is increased, it occurs at a finite bond-dimension independent crossover temperature in the latter.