There is no long-range order in the relevant case [1] (the notion of correlation length is hereby undefined, it being based on an exponential decay, and not an algebraic one); there is a topological order, as discussed by Kosterlitz and Thouless [2]. This is associated with the logarithmic divergence of the variance of the phase of the order parameter for increasing distance. The underlying mechanism is considered in detail in the two celebrated papers by Berezinskii [3,4]. Briefly, in two-dimensions, the integrand of the integral representation of the variance of the order parameter decays insufficiently fast (its amplitude decays like 1/||k||) for ||k|| → ∞, leading to the above-mentioned logarithimic divergence of the variance (a simplified discussion of this issue is presented in Sec. 11.2, beginning on p. 345, of Ref. [5]).

[1] ND Mermin, H Wagner, Phys. Rev. Lett. 17, 1133 (1966).

[2] JM Kosterlitz, and DJ Thouless, J. Phys. C 6, 1181 (1973).

[3] VL Berezinskii, Sov. Phys. JETP 32, 493 (1971). [Classical Systems]

[4] VL Berezinskii, Sov. Phys.JETP 34, 610 (1972). [Quantum Systems]

[5] N Goldenfeld, Lectures on Phase Transition and the Renormalization Group (Perseus, Reading, MA, 1992).

I would slightly differ from the preceding post. We do have a well-defined correlation length. To define it, we must describe the transition in detail. In a picture where we consider vortices as well defined excitations (which is reasonable, since they can be unequivocally and uniquely identified in any configuration) the transition is one between an ``insulating'' phase, where we only have dipoles of vortices, namely bound pairs of dipoles of opposite sign, and an conducting phase, in which there are free vortices. An infinitesimal preference for, say, positive vortices will yield a ``charged'' phase, with a net density of positive vortices.

In both these phases we may define a coherence length. In the insulating phase it corresponds to the length of the largest typical dipole, in the conducting phase it is the distance between free charges (both concepts are a bit tricky, since it is not unambiguous to assign to each vortex its companion in the dipole which it forms, nor is it obvious to distinguish a free charge from a bound one). In the conducting phase, you can also define the coherence length via the screening length, which gives the exponential decay of the electric field around a charge held fixed at the origin.

That the coherence length, thus defined, grows as a stretched exponential, is a fact. I am not aware of a simple proof of it.

Dear F. Leyvraz, I did not say that there were no phase transition (already the designation Kosterlitz-Thouless transition conveys the fact that there is a transition), but that the notion of correlation length does not apply in the case at hand -- the transition is not characterised by a correlation length ξ, describing the decay of the form exp(-r/ξ) of a relevant correlation function for increasing r, that would diverge on approaching the critical point. In the case at hand, the Mermin-Wagner theorem applies for T ≠ 0: there is no long-range order at non-vanishing temperatures.

Could you please clarify your opening statement?

I said I differed slightly: precisely because there is, in my opinion, a well-defined correlation length, though it is not the one corresponding to the decay of the two-spin correlation function. More precisely, the correlation length, defined in the vortex picture as the screening length, does in the high-temperature phase coincide with the length characteristic of exponential decay of the two-spin correlation function. On the other hand, the correlation length in the low-temperature phase, defined as the upper limit of the size of dipole (or generally speaking, neutral aggregate) of vortices, has no such obvious description in terms of the two point correlation decay, at least to my knowledge.

I have edited my first post to make my point clearer.

Dear F. Leyvraz, since we were discussing, I would rather have that you did not edit your original post, since that detracts from my above response; the result of editing is as if I were writing to myself.

Dear Dr. Farid: for the record, let me then state that the original post stated: ``I differed slightly from the previous post: There does exist a true phase transition: if we look consider  vortices as well defined excitations...''

I did not think about the effect of my editing on the meaning of your post, for which I apologise. However, it seemed to me that, for the rest of the readership, a clearer post is to be preferred, hence I prefer to stick to the edited version of the post.

Thank you Dr Leyvraz. The emphasis 'There does exist a true phase transition' (note the emphatic word 'does') unequivocally implies that I had denied the existence of the Kosterlitz-Thouless phase transition, which I had demonstrably not. I had indicated that the notion of coherence length is not applicable in the absence of a phase transition characterised by a long-range order.

Surely you are right. My post was carelessly written in that respect, but I maintain my basic point: there is a meaningful correlation length, which in the high temperature phase is  linked to the exponential decay of the two-spin correlation function, and which does increase exponentially, more accurately, as a stretched exponential, as T_c is approached. The significance of this quantity is best appreciated in the Coulomb gas language, where the transition itself is also easier to understand.

That was what I meant when I said that I differed (slightly) from your post.

Dear Dr Leyvraz,

My second comment on this page was not about the contents of your first comment in its entirety that you describe, but about the fact that your emphatic statement implied that I had somehow denied the existence of the Kosterlitz-Thouless transition, which I had not.

As for the high-temperature phase, at high temperatures there is no order due to the complete dominance of thermal fluctuations (in turn due to the entropic contribution to the free energy), so that while correlation functions are exponentially decaying functions of distance, this behaviour is distinct from that close to a critical temperature Tc  where the temperature dependence of various quantities are described in powers (including fractional powers and logarithmic corrections) of (T-Tc) and (Tc-T), depending on whether T > Tc or T < Tc. In the Landau theory of second-order phase transitions, various expansions are centred around Tc, where one knows (by definition) that the order parameter can be made as small as desired by choosing T sufficiently close to Tc (even then, Landau's theory does not apply for T  very close to Tc, because of critical fluctuations, but this region is often too small to be relevant to some experiments). It is here where one considers the order parameter and the attendant divergence of the correlation length. In this light, I had not in mind a finite correlation length in the high-temperature phase when citing and referring to the Mermin-Wagner theorem. I was very focussed on a finite temperature region centred on Tc.

We agree that the Landau theory does not apply to the KT phase transition. My point is the following: in that transition, the 2-spin correlation function goes from an algebraic decay below T_c, to an exponential decay above T_c. This exponential decay is characterised by a decay length, which I may call xi, and which indeed diverges at T_c, indeed, it does so exponentially in |T-T_c|^(-1/2) if memory serves me right, which is the very object of the question.

Of course, in the low-temperature phase, there is no length characteristic for the algebraic decay of the 2-spin correlation function. However, in the Coulomb gas picture, I am quite sure it is possible to introduce such a length, namely the one where the running dielectric constant converges to its macroscopic value, which I had earlier denominated the ``size of the largest dipole''. This, of course, also diverges at T_c.

Finally, the Mermin-Wagner theorem merely states that there never can be a non-zero magnetisation in the 2d XY model (as opposed to the 3d one). This is true for all T, and is therefore not very illuminating if we ask what the nature of the transition is, or how the correlation length is defined. It is correct that the MW theorem was long thought to mean that no phase transitions occur in 2d, but we now know this interpretation to be wrong: the type of transition we have involve transitions in the 2-spin correlation function: while it always decays to zero, by the MW theorem, it may switch from doing so algebraically to doing so exponentially. This is what happens in the XY model, where the transition is driven by the appearance of free vortices.

I assume we may agree on these issues.

The specific correlation length to which you refer in the first paragraph of your latest writing, was first calculated by Kosterlitz (J. Phys. C 7, 1046 (1974)), however it is not a correlation length corresponding to any correlation function, but is a result of an identification in the underlying RG calculations. The spin-spin correlation length that he in the same paper explicitly calculates corresponds to T↓Tc, or t  → 0+, showing the spin-spin correlation function Γ(r) to decay as a power-law, like 1/r¼, for r → ∞.

Incidentally, the Mermin-Wagner theorem applies to more general systems than the XY model. It is tied to the spontaneous breaking of a continuous symmetry and the attendant gapless Goldstone modes. See, for instance, Mermin (Phys. Rev. 176, 250 (1968)),*) Hohenberg (Phys. Rev. 158, 383 (1976)), and Brézin and Zinn-Justin (Phys. Rev. B 14, 3110 (1976)).

*) The original Mermin-Wagner paper is dated 1966.

I am puzzled by your statement that at T>T_c the decay should be algebraic. Is that really what you meant? To the best of my knowledge, it is exponential. And such a decay  does define a correlation length. That, once you reach T_c, the decay becomes algebraic is, of course, unsurprising.

I am quite aware of the Mermin-Wagner theorem, and it extent. For other models, however, such as the Heisenberg model in 2d , there indeed seems to be no transition. On the other hand, for the ``liquid-solid'' transition, the situation is similar, though yet more complicated: the MW theorem states that sharp Bragg peaks cannot occur.

Nevertheless,  precisely because the theorem has such a wide range of applications in models which sometimes do, and sometimes do not, have a KT transition, I do not understand very well the reason for your insisting on its importance.

To be precise, while Kosterliz calculates Γ(r), the spin-spin correlation function, for t = 0+, the expression I presented corresponds to the limit t = 0, the limit being thus taken from above. You can verify the expression by consulting the relevant paper by Kosterliz, the details of which I presented in my previous writing (see Eq. (4.10) on page 1053).

Regarding the Mermin-Wagner theorem, I did not make any statement regarding its importance, or otherwise. In my very first comment I merely indicated that the explicit calculations by Berezinskii established the divergence of the variance of the phase of an order parameter (a complex field) in two dimensions, in conformity with the Mermin-Wagner theorem. That is all. Mermin, and Mermin and Wagner are cited by both Brezinskii and Kosterlitz and Thouless in their relevant papers.

An important paper, mostly overlooked, is by Wagner (Z. Physik A 195, 273 (1966)), received on 26 April 1966; the Mermin-Wagner paper was received on 17 October 1966.

I have checked the paper by Kosterlitz which you mention. The computation leading to (4.10) is for the low temperature regime, as follows from the sentence immediately preceding (4.6) whereas the high temperature behaviour of the correlation function is given in (4.13) and involves, apart from the power-law, a rapid decay on the scale xi, where xi, compare (3.11) of the paper, does indeed diverge exponentially.

No! The lower-case t as usual stands for T - Tc, or the dimensionless quantity (T - Tc)/Tc, so that t > 0 is the same as T > Tc, and t < 0 the same as T < Tc. See for instance the text preceding Eq. (1.8), on p. 1047 (clearly, otherwise the t < 0 in Eq. (1.7), p. 1047, would not make sense).

I think we should close this discussion here, since it has descended into arguing about symbols.

I attach the link to the relevant paper by Kosterlitz below for the interested.

https://doi.org/10.1088/0022-3719/7/6/005

https://doi.org/10.1088/0022-3719/7/6/005

I would merely recommend Zhe Wang, and any reader of this interchange, to take advantage of the link kindly  provided by Dr. Farid and make up his/her own mind as to what the computation means after reading carefully the text from after equation (4.5) to the end of the section. It is not necessary to understand the derivations of the equations. I do not expect, at this level, to convince Dr. Farid.

Merely for the record, I am not arguing about symbols: just before (4.6) Kosterlitz says:

``Let us first consider the correlation function below the critical temperature.''

and nothing between this and (4.10) cited by Dr. Farid closes this remark. On the other hand, the second paragraph after eq (4.11) begins by

``The situation for t>0 is similar to that of the previous section in our discussion of the correlation length''

As Dr. Farid correctly remarks, the high temperature phase is, indeed, t>0. After this sentence comes, among other things, eq. (4.13), which in my opinion describes the high-temperature behaviour.

Wherewith I am glad to close this discussion.

Dear Dr Leyvraz,

You mention "I do not expect, at this level, to convince Dr. Farid." Could you please clearly specify to convince me of what precisely? The computations by Kosterlitz are unequivocal and I cannot make out what in your latest comment is contrary to what I have mentioned earlier.  The paper by Kosterlitz was notably introduced by me on this page!

Dear professors,

Thank you so very much for the in-depth discussions! 

I met prof. Kosterlitz today and consulted with him on this question.

According to him, there two distinct correlations involved in the KT transition,

1. the correlation length xi, which may be interpreted as the mean separation between topological defects decays exponentially above the criticality t>0, and is infinite below t

You are welcome Zhe.

Regarding the high-temperature correlation length, which is finite, as I indicated in one of my earliest comments on this page, it is purely a thermal effect in the disordered phase, unrelated to the KT transition. It was calculated in the 1960s (by Stanley, Kaplan, Moore, and others), using cluster calculations. Calculation of the leading-order contribution to the spin-spin correlation function of the XY model on a square lattice (in d = 2) at high temperatures has been detailed in the book by Altland and Simons (Condensed Matter Field Theory, 2nd edition, (Cambridge University Press, 2010)),  Sec. 8.6, beginning on p. 463. Actually, the expression for the high-temperature correlation length can be deduced without any calculation; combining the parameters of the model in such a way as to produce a quantity with the dimensionality of length, one arrives at the expression for this correlation length..

Dear Prof. Farid,

Thank you for the reply!

Regarding the high-temperature correlation length xi, please, could you kindly explain a little bit on the physical intuition of the factor (-1/2) in the following expression:

xi=exp[a*t^(-1/2)] ,where a is a constant.

In fact, I have consulted quite a few professors on the past week but without a satisfactory. 

Thank you.

Very kind wishes,

Wang Zhe 

You are welcome Zhe.

The factor -½ follows from the scaling relations deduced by Kosterlitz in J. Phys. C 7,1046 (1974), which are akin to the scaling relations for the Coulomb gas deduced by Anderson and Yuval (Phys. Rev. Lett. 23, 89 (1969)), Anderson, Yuval, and Hamann (Phys. Rev. B 1, 4464 (1970)), and Anderson (J. Phys. C 3, 2436 (1970)); the first two are cited as references [1] and [2] in the paper by Kosterlitz. Now, if you consider Eq. (3.4) in the paper by Kosterlitz, and take into account that at the relevant fixed point x ≡ βp2 -2  = 0 (see the text preceding Eq. (3.1)), it follows that at this fixed point -y2 = x02. Since x02  =  -Ct, with C ≈ 2.1 (see the text following Eq. (3.4)), it follows that at the relevant fixed point y, that is y0, is equal to (Ct)½. The factor -½ follows from using the latter result in the middle expression in Eq. (3.11).  In this connection, note that b ≡ π/C ≈ π/2.1 = 1.49599... ≈ 1.5.  □

Ps. Actually, b ≡ π/C½ ≈ π/√2.1 = 2.1679... ≈ 2.17.