The classical twodimensional 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 $O(3)$ nonlinear sigma model in $1+1$ dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higherdimensional ones (like quantum chromodynamics in $3+1$ dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finitetemperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of stateoftheart 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 twodimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.06310
 Bibcode:
 2021arXiv210606310S
 Keywords:

 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Lattice;
 Quantum Physics
 EPrint:
 28 pages, 10 figures, slightly updated data and improved presentation of scaling procedure