The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

Abstract

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+11+1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+13+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)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 \chi_E^\text{eff} \sim 1500χEeff∼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-TT transition and asymptotic freedom, though with a slight preference for the second.

Contrasting pseudocriticality in the classical two-dimensional Heisenberg and RP2 models: Zero-temperature phase transition versus finite-temperature crossover

“…We note that our conclusions here are largely consistent with those of Ref. [20], where the results of an analysis based on similar tensor-network methods leaned more towards the scenario of asymptotic freedom than that of a finite-temperature transition.…”

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

“…In particular, our results show no sign of a possible BKT-like divergence around a temperature T ≈ 0.5 as was recently suggested in Ref. 20, where similar tensor-network methods were used. When considering extrapolated correlation lengths as opposed to raw MPS correlation lengths, which can be notoriously inaccurate even for large bond dimensions [18], we observe no signature of such a divergence.…”

“…A similar approach was pursued for the case of the Heisenberg model in Ref. 20 where hints of a BKT transition were reported, but our results do not seem to validate this conclusion.…”

Topological magnetic phase transition in Eu-based A -type antiferromagnets

“…This model has a continuous rotational symmetry, which, according to the Mermin-Wagner Theorem [1], prevents the spontaneous breaking of this symmetry by the formation of infrared divergent Goldstone modes, i.e., spin waves. While this system has no symmetry-breaking phase transition, the presence or absence of a phase transition in the 2d classical Heisenberg ferromagnet has remained an open and controversial question [34][35][36]. Monte Carlo studies, as in the present work, are obligated to introduce a finite size…”