The norms or expectation values of infinite projected entangled-pair states (PEPS) cannot be computed exactly, and approximation algorithms have to be applied. In the last years, many efficient algorithms have been devised-the corner transfer matrix renormalization group (CTMRG) and variational uniform matrix product state (VUMPS) algorithm are the most common-but it remains unclear whether they always lead to the same results. In this paper, we identify a subclass of PEPS for which we can reformulate the contraction as a variational problem that is algorithm independent. We use this variational feature to assess and compare the accuracy of CTMRG and VUMPS contractions. Moreover, we devise a new variational contraction scheme, which we can extend to compute general N-point correlation functions.

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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 (MPSs) 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 finitetemperature phase transition, supporting the scenario of asymptotic freedom. For the RP 2 model we observe an abrupt onset of scaling behavior, 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 pseudocritical 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 behavior 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 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.