It is not true. There is the celebrated Onsager solution for the phase transition of a 2-dimensional Ising model. What you probably refer to is the absence of a phase transition in a 2 dimensional Heisenberg system. This is a deep statement known as the Mermin Wagner theorem. Roughly speaking it is a consequence of the necessary existence of a Goldstone mode in a system with rotational symmetry such as the Heisenberg system. In such a system there must be an excitation with zero energy in the limit of long wavelength (same concept that gives rise to the Higgs boson). From this one can show that there can't be spontaneous magnetization in a Heisenberg system of less than 3 spatial dimensions. The spin wave approximation gives rise to divergent integral expressions. sing the spin wave approximation as an argument is not rigorous, but an intuitive approach to the Mermin Wagner theorem.

Let me just add a brief comment to Christian's answer. He refers to the statement that for two-dimensional systems with a continuous order parameter and sufficiently short range interactions the Mermin-Wagner theorem states that there cannot be any long-range order. This is not to be confused with a phase transition, though, because we know of examples where such systems undergo phase transitions without breaking their continuous symmetry. The best-known example is the Kosterlitz-Thouless transition in the classical XY model, which amounts to a vortex binding/unbinding transition that does not induce overall long range order.

I would like to add that systems of self-propelled particles moving in two dimensions and interacting through short range alignment forces, exhibit a phase transition where the ordered phase is characterized by long range order. This novel phase transition was reported in 1995 in a paper by Vicsek and collaborators. See for instance

http://arxiv.org/abs/cond-mat/0611742.

There are other ways to get around Mermin-Wagner even allowing some long-range order, but not breaking of rotation symmetry.

Just as Mermin-Wagner this is all rigorous stuff.

Two examples below.

Phase transitions for two-dimensional models with isotropic short-range interactions and continuous symmetries

SB Shlosman

Communications in Mathematical Physics 71 (2), 207-212

First-order transitions for n-vector models in two and more dimensions; rigorous proof

Authors

Aernout CD Van Enter, Senya B Shlosman

Publication date

2002/5/22

Journal name PRL 89, 285702

arXiv preprint cond-mat/0205455

I am sorry that I have to tell you that the magnetic ordering transition is not executed by local Heisenberg interactions but by a boson field. In RG theory the boson driven phase transition has been called a stable fixed point (SFP). The main issue of RG theory was that spins and interactions between spins are unimportant for the dynamics in the critical range. As a consequence, the critical exponents are independent of spin structure (universal). In all magnets with three-dimensional spin the ordering transition is of the SFP type. Exceptions are genuine Ising magnets that exhibit atomistic dynamics. The field bosons are essentially magnetic dipole radiation emitted upon precession of the magnetic moments. The simple experimental criterion to identify the field dynamics is the finite widths of the critical range. In atomistic models such as the Heisenberg model the critical power functions hold asymptotically at the critical point only. The same applies to T=0. Bloch´s solution ( the T^3/2 function) holds asymptotically only. Onsagers critical power function with exponent 1/8 is the first term of a power series expanion at Tc. Since the spin dynamics comes from the boson guiding field only field theories can give correct description. However, the field quanta are not precisely known as yet. Realistic field theories therefore are not yet possibel.

Dear Ali,

from a theoretical viewpoint, not only two-dimensional phase transitions exist (for discrete order parameters), but furthermore there are exact analytical solutions for their study. For a class of (so-called minimal) models including percolation, Ising (magnets and fluids), Potts, Z_n, they can be studied with conformal field theory. See for example the book by Mussardo G., "Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics". One of the unsolved problems in statistical mechanics (and theoretical physics) is to find analytical solutions for three dimensional models exhibiting phase transitions.

Best regards,

daniele

It  is rather,   when    you think of a phase transition (continuous) you always think of  Ising transition, which is solved exactly  in 2-spatial dimension.

Quantum phase transition  can  occurs in 1D   [ a consequence of  " (d+1)-dimensional stat mech is equivalent  to   D-dimensional  quantum mech]

And , what you refer    that continuous  sysmmetry can not be broken in  2D  is called   Mermin Wagner theorem.   See  wiki  http://en.wikipedia.org/wiki/Mermin%E2%80%93Wagner_theorem

Look at " Chaikin and Lubensky",  a good book.

First of all, what is meant by dimensionality? In atomistic models it is distinguished between the dimensionality of spin and the dimensionality of exchange interactions. Because spins and exchange interactions between spins are not relevant for the dynamics (of the spins!) these dimensionalities are unimportant. As a consequence of RG theory (K.G. Wilson) ordering transitions in magnets with three-dimensional spin are executed by a boson field. The boson field is a vector field and also has a dimension. This is the relevant dimension in ordered magnets. The finding of RG theory that the magnetic ordering transition is executed by a boson field has two dramatic consequence, first that the Heisenberg model is not realized in nature and, second that the Mermin-Wagner thorem is inapplicable. FeCl2 and K2NiF4 are materials with exactly two-dimensional magnon dispersions and high ordering temperatures. Boson dynamics is easily recognized by the finite critical range. Note that the critical power functions of the atomistic models hold asymptotically at critical point only.

The only exceptions are genuine Ising magnets. In Ising magnets the dynamics is as predicted by atomistic models. K2CoF4 and Rb2CoF4 are the only known examples (Ikeda, Hirakawa, SSC). Other genuine Ising magnets do perhaps not exist. It is a very fortunate fact that at least Onsager´s atomistic 2D Ising model is realized in nature.

I assume the conclusion is that for the Heisenberg model (i) the Mermin-Wagner theorem holds, but (ii) there still exists a Kosterlitz-Thoughless type phase transition. Is this correct? The body of evidence for a phase transition seems to be a bit at odds with the statement of the Mermin-Wagner theorem and begs for an explanation.

No, in the nonlinear Heisenberg model I studied with Shlosman, there exists a first-order transition between two paramagnetic, nonmagnetised phases with different correlation lengths. So in this example I) is correct and II) is not

Thanks so much! But I am referring to the plain vanilla nearest-neighbor Heisenberg model.

The conjecture is that the plain vanilla (nearest-neighbour, 3-component ferromagnetic )Heisenberg model with interaction the inner product between neighbouring spins has no phase transition. There have been defenders of the minority point of view that is happens, especially Patrascioiu and Seiler .

But our model which had a different form of nearest-neighbour interaction (high power of the inner product) satisfied Mermin-Wagner, had no Kosterlitz-Thouless phase, was ferromagnetic, indicates that the absence of a phase transition in some sense is more of an accident and does not follow from general principles. I would not therefore say it is "impossible", but just that it does not happen to occur. But this may just be linguistic.

One should distingush 2-dim statistical systems and (1+1)-dimensional field theory. For the latter, the impossibility to break

spontaneously a continuous symmetry follows from the Coleman theorem - the absence of interacting massless Goldstone particles

[S. Coleman, Commun. Math. Phys., {\bf 31} 259 (1973)]. A discrete symmetry can be broken. But it is necessarily restored

for any nonzero temperature. Indeed, the physical picture of spontaneous breaking of discrete symmetry imply the presence

of domain walls between the different phases. But if you have only one spatial dimension, these domain walls are simply

massive particles. For any nonzero temperature, they are present in the heat bath. And that means that the system cannot

stay in one phase and the order parameter is zero. In fact, the physics of (1+1) dimensional field theories is the same as the physics

of 1-dimensional statistical systems. In 1-dimensional Ising model, the Z_2 symmetry can be broken at zeto temperature, but it is

restored under heating, however small the temperature might be.

This is actually a good reference:Article Theory of Phase transitions in two-dimensional sys- tems,