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

Abstract

Tensor-network methods are used to perform a comparative study of the two-dimensional classical Heisenberg and RP 2 models. We demonstrate that uniform matrix product states (MPS) with explicit SO(3) symmetry can probe correlation lengths up to 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 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 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.

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

“…Recently, we learned that Burgelman, Vanderstraeten, and Verstraete have also studied the Heisenberg and RP 2 models using tensor-network method and reached a similar conclusion [61]. While our results seem largely consistent with theirs, it would be interesting to make detailed comparisons.…”

Critical correlations of Ising and Yang-Lee critical points from Tensor RG

“…The latter, of course suffer from their own problems inher- ent in sampling large lattices, including but not restricted to critical slowing down, all of which are entirely absent here. While finite bond dimension can be related to finite length rounding of critical correlations in variational MPS based approaches to classical criticality 26 , the exisence of such finite length rounding is far from obvious in TRG and other multiplicative renormalization approaches. Thus it is somewhat unclear (to us) how to relate technical errors that are clearly present in the computation to the physical errors (if any) in conclusions drawn.…”