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.
Highlights
Accurate finite-size scaling, seemed rather to suggest a pseudo-critical region [19] and quasilong-range order [20]
The most striking feature of the latter is the non-perturbative phenomenon of asymptotic freedom [24, 25]: Namely the effective coupling constant being vanishing at large energies, while tending to get stronger and stronger at low energies, to what happens in the celebrated case of 3+1-dimensional quantum chromodynamics (QCD) [26,27,28]
It is fair to say that the specifics of the phase transition of the classical Heisenberg model in two spatial dimensions has remained inconclusive despite all these attempts
Summary
The classical Heisenberg model represents the workhorse of statistical mechanics for capturing (anti-)ferromagnetism when no lattice distortion or any other anisotropy source plays a major role. This is a topological invariant characterizing the configurations of n(x, τ) in spacetime, i.e., the homotopy classes via π2(S2) = Such a θ -term (often dubbed Weiss-Zumino action term) is the one responsible for the peculiarly different behaviour, gapped vs gapless, of integer vs half-integer spin Heisenberg chains in d = 1, at the heart of the celebrated Haldane conjecture [64]. Enough, such a topological term is the one generated in the semi-classical context by certain lattice distortions and plays a central role in the description of Skyrmions (i.e., the defects with |Q| = 1) in real materials [65,66,67,68]. In the absence of an explicit coupling, there exist quite compelling arguments to rule out such Skyrmions from playing the role of vortices in O(2) theories, i.e., being responsible for topological transitions à la BKT [69]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
