Contrasting pseudocriticality in the classical two-dimensional Heisenberg and RP2 models: Zero-temperature phase transition versus finite-temperature crossover
Lander Burgelman, Lukas Devos, Bram Vanhecke, Frank Verstraete, Laurens Vanderstraeten
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 finite-temperature 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 (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 finite-temperature 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.
We study the classical two-dimensional RP^{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 due 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 RP^{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 RP^{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 RP^{2} models. Moreover, there seems to be no appropriate candidate of conformal field theory matching the spectrum, which shows the effective central charge c∼1.9. These suggest that both models have a single disordered phase at finite temperatures, although the ferromagnetic RP^{2} model exhibits a strong crossover at the temperature where the dissociation of Z_{2} vortices has been reported.
We apply variational tensor-network methods for simulating the Kosterlitz-Thouless phase transition in the classical two-dimensional XY model. In particular, using uniform matrix product states (MPS) with non-Abelian O(2) symmetry, we compute the universal drop in the spin stiffness at the critical point. In the critical low-temperature regime, we focus on the MPS entanglement spectrum to characterize the Luttinger-liquid phase. In the high-temperature phase, we confirm the exponential divergence of the correlation length and estimate the critical temperature with high precision. Our MPS approach can be used to study generic two-dimensional phase transitions with continuous symmetries.
The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to L=16384. Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with L. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.
Abstract The quantum Gaussian phase transition is investigated for the infinite spin-1 bond-alternative Heisenberg model in one spatial dimension. By using a tensor network representation with an infinite matrix product state approach, the ground state energy, bipartite entanglement entropy, non-local string order, and fidelity per lattice site are calculated to characterize the phase transition. At the quantum phase transition point, the scaling behavior of various physical observables with respect to the finite truncation dimension are discussed for the ground state wavefunctions. In addition, the central charge is extracted from the finite entanglement entropies and the finite correlation lengths. Furthermore, the various critical exponents of the string order are calculated. The characteristic critical exponents and the central charge determine the universality class of the phase transition.
The classical two-dimensional Heisenberg model revisited: An SU(2)-symmetric tensor network study
The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3) O ( 3 ) non-linear sigma model in 1+1 1 + 1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+1 3 + 1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU(2) S U ( 2 ) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to \chi_E^\text{eff} \sim 1500 χ E eff ∼ 1500 , a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite- T T transition and asymptotic freedom, though with a slight preference for the second.
The classical two-dimensional Heisenberg model revisited: An SU(2)-symmetric tensor network study
The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3) non-linear sigma model in 1+1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU(2) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to χEeff∼1500, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-T transition and asymptotic freedom, though with a slight preference for the second.