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

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 (MPSs) with explicit $\mathrm{SO}\left(3\right)$ symmetry can probe correlation lengths up to $\mathrm{O}\left({10}^3\right)$ 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 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 ${\mathrm{RP}}^2$ 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.

Figure 1

The (a) free energy density, (b) energy per link, and (c) specific heat as a function of temperature in the Heisenberg model. A comparison to the Monte Carlo results of [71] was added in (b). (d) Convergence of the energy per link with MPS truncation error at $T=0.55$

Figure 2

The (a) free energy density, (b) energy per link, and (c) specific heat as a function of temperature in the ${\mathrm{RP}}^2$ model. A comparison with the results of [42] was added in (b) and (c), depicting the data obtained using both Boltzmann sampling (B) and equilibrium ensembles constructed from the density of states (RW). (d) Convergence of the energy per link with MPS truncation error at $T=0.35$.

Figure 3

Illustration of the correlation length extrapolation Eq. (14) for the ${\mathrm{RP}}^2$ model at $T=0.36$. While the lowest bond dimensions clearly fall outside of the proper scaling regime, the inset shows that restricting to the 10 largest bond dimensions yields a particularly clean result of $\xi =188\pm 1$.

Figure 4

Extrapolated correlation length as a function of temperature for the (a) Heisenberg and (b) ${\mathrm{RP}}^2$ models.

Figure 5

BKT-type fit (16) to the correlation length divergence in the ${\mathrm{RP}}^2$ model, giving an estimate $T_c=0.339\pm 0.001$.

Figure 6

Scaling of the entanglement entropy as a function of correlation length in the ${\mathrm{RP}}^2$ model for temperatures (a) $T=0.4$ and (b) $T=0.3$. The dashed line in (a) represents the extrapolated correlation length at this temperature. The line in (b) corresponds to a fit of the scaling form (17).

Figure 7

Effective central charge as a function of temperature in the (a) Heisenberg model and the (b) ${\mathrm{RP}}^2$ model.

Figure 8

Entanglement spectrum of a fixed point MPS with $D=4780$ for the ${\mathrm{RP}}^2$ transfer matrix at $T=0.3$. The dashed line shows a quadratic envelope of the form $a\ell (\ell +1)+b$.

Figure 9

Entanglement spectrum from Fig. 8 split into (a) even and (b) odd integer spin sectors. After a shift of each sector followed by a global rescaling, we obtain equidistant spectra for even and odd integer spin sectors, respectively. The dashed lines indicate approximate energy levels.

Figure 10

Entanglement spectra of fixed points for the Heisenberg model at $T=0.6$. (a) Spectra of fixed point MPSs (left) without and (right) with imposing $\mathrm{SO}\left(3\right)$ symmetry. The different integer spin sectors in the symmetric spectrum are plotted in different colors. (b) Spectrum for a fixed point MPS with half-integer spins on the virtual level.

Figure 11

Comparison of results for the ${\mathrm{RP}}^2$ model with ${\ell }_{\text{max}}=6$ and ${\ell }_{\text{max}}=10$. (a), (b) The extrapolated correlation length and effective central charge as a function of temperature. (c), (d) The relative difference in the free energy density and the energy per link between both cutoffs as a function of temperature.

Figure 12

Results for the Heisenberg model with ${\ell }_{\text{max}}=2$ and a maximal MPS bond dimension $D=3000$, showing (a) the extrapolated correlation length and (b) the effective central charge as a function of temperature.

Figure 13

Results for the ${\mathrm{RP}}^2$ model with ${\ell }_{\text{max}}=2$ and a maximal MPS bond dimension $D=3000$. (a) The extrapolated correlation length as a function of temperature. (b) A fit of the BKT scaling form (16) to the correlation length divergence. (c) The effective central charge as a function of the temperature. (d) The entanglement-entropy scaling at $T=0.15$.