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.
Thermal tensor network simulations of the Heisenberg model on the Bethe lattice