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 $O(3)$ 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 ${\chi }_E^{\text{eff}}\sim 1500$, 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.

arXiv comment28 pages, 10 figures, slightly updated data and improved presentation of scaling procedure

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• Tensor Network Renormalization Study on the Crossover in Classical Heisenberg and $\mathrm{R}{\mathrm{P}}^2$ Models in Two Dimensions
• Ground state properties of the Heisenberg-compass model on the square lattice
• Asymptotic Freedom and Finite-size Scaling of Two-dimensional Classical Heisenberg Model
• Efficient optimization of variational tensor-network approach to three-dimensional statistical systems
• Characterizing spin ordering via maximal row correlation in classical spin models
• Fluctuations of topological charges in two-dimensional classical Heisenberg model
• Nonpertubative Many-Body Theory for the Two-Dimensional Hubbard Model at Low Temperature: From Weak to Strong Coupling Regimes