The Mermin-Wagner theorem implies for the standard classical Heisenberg model in two spatial dimensions that there cannot be spontaneous symmetry breaking. At the same time, there is a body of evidence, often also from the 60ies, that the system is undergoing a phase transition at finite temperatures. How is this compatible? Is it that this is a Kosterlitz-Thoughless type phase transition? Where is the catch?
Yes, that is a BKT transition.
for classical O(2) Rotor(XY) model, the energy cost to separate a vortex and an anti-vortex energy is ~ln(L). (vortex confinement energy)
The entropy of a vortex (location uncertainty) ~ln(L^2)=2ln(L), which is of the same scale as the vortex confinement energy.
F=E-TS, there should be a critical T (BKT point) where the vortices tend to proliferate in space and disorder the phase.
at low T, there is no symmetry breaking due to M-W theorem. However, M-W only excludes long-range order. The classical rotor at low T has quasi-long-range order with a power-law-decay correlation.
actually, in the case of a discrete lattice of spins, two-dimensional XY model can be evaluated using the transfer matrix approach and thus map to the 1d quantum rotor(XXZ) model at zero temperature.
There we can clearly visualize that the low T regime corresponds to the gapless XY phase(1d quantum XY model) where the boson compactification is irrelevant(which means the vertex term would not proliferate) and the GS has power-law decay correlation.
Upon such mapping, the 'T' in classical 2D BKT transition corresponds to the Luttinger parameter `K' for 1d quantum rotor model. If K is small, the vertex operator (regarded as a 1+1 d space time instanton akin to 2d vortex) becomes relevant and tends to proliferate in space time. This would gap out the rotor and reach a Mott phase.
Wonderful, thanks so very much for your comprehensive answer. Yes, what you write is compatible with what I know. Just to be sure, all you say also applies to the Heisenberg model and not only the two-dimensional XY model, right?
Sorry, I probably use both languages interchangeably and make it a mess...
My previous answer applies to 2D classical O(2) Heisenberg (or called O(2) rotor, XY). That is where BKT transition appears. In particular, the system only exhibit a U(1) symmetry. There we expect a vortex like defect in 2D(which is akin to 1+1 d instanton in quantum system) and the vortex proliferation drives the BKT transition.
if you are considering classical O(n) Heisenberg with n>2, that is a separate story.
e.g, n=3, the O(3) vector does not render a vortex like defect in 2d and the vortex proliferation argument should not apply.
the current understanding is, for O(3) classical Heisenberg model in 2d, there is no finite T transition as interaction of Goldstone particles in two dimensions lead to the infrared catastrophe.
(Article Interaction of goldstone particles in two dimensions. Applic...
)some analytical result:
1) Polyakov et.al implement a dimension expansion(2+\epsilon dimension) + RG calculation and show that for d=2(\epsilon=0), the critical T is zero for O(n) NLSM with n>2.
2) I am not a numerical person. I heard there are some MC studies showing a critical temperature(at very low T) for 2d O(3) Heisenberg model. however, current understanding is that it is a pseudo-critical behaviors due to finite size.(the correlation length is finite, but as large as 500 sites >> system size)
Probably the actual Heisenberg model, with 3d spins and short range interactions
does not have a phase transition in two dimensions. See: Article Phase Transition in a Two-Dimensional Heisenberg Model
Certainly, Henk van Beijeren is correct. In the standard 2d Heisenberg model, with
nearest-neighbor interactions, there is no phase phase transition, in contrast to the 2d XY model with its BKT transition. It may be interesting to note that, for the 2d Heisenberg model, many years ago, based on an incorrect interpretation of data from a high temperature expansion, a phase transition of BKT-type had been predicted. The existence of a conventional transition with a non-vanishing order parameter, at low temperatures, had been ruled out, already even earlier, by the famous "Mermin-Wagner theorem". The erroneously predicted "transition" in the Heisenberg model had attracted some attention, and it had been even given a special name.
The crucial argument, ruling out a phase transition, of conventional or non-conventional type, in the 2d Heisenberg model has been put forward by Alexander Polyakov in Physics Letters B59, 79 (1975).
Attention may be also drawn to the article "Spin-ordering in a planar classical Heisenberg model" by Franz Wegner in Z .Physik 206, 465 (1967). There, for d=2, in the XY model the spin correlations are shown to decay algebraically at low temperatures, ruling out a conventionally ordered low-T phase, but allowing for a
phase transition to an exponentially decaying high-T phase. These results have been appreciated in the famous work by Kosterlitz and Thouless in 1973, emphasizing the role of vortices for the B(erezinskii)-K(osterlitz)-T(houless) transition. By the way, Vadim Berezinskii published his pioneering research for d=2 in 1971, and he died already in 1980.
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