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与Perk教授及夫人的学术讨论的e-mails-4
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本期公开Perk教授4月份的五封来信和我的回复。为了集中于学术讨论，本人将屏蔽掉e-mails中的非科学内容（用……表示）。为了正确完整地反应交流中的学术思想，将不对e-mails做任何翻译 [注：与Perk教授及夫人的学术讨论的e-mails就暂时公开到这里，以后的讨论如果没有新的重要内容将不再公开] 。
Perk教授的第9封来信(2009-4-1)：
Dear Zhidong,
References [9] and [10] are not easy to read, so that I wrote a new derivation only using Suzuki in my addition. However, the conclusion can be extracted also from [9] and [10].
In [9] there is fine print talking about beta=0, apart from also citing [10] (or their [12]). One of the things to note is B(beta) in (9), which for Ising becomes |exp(beta J)-1|, going to zero as beta goes to zero. Analyticity of beta f as a function of beta follows then for beta small enough (as long as H is real finite), after carefully interpreting what is said.
In [10] you are not to set beta=0, since you are going to estimate the radius of convergence. The identities are true for beta=0 with RHS=0 or RHS=1/0=infinity, but that does say nothing about estimating the nonzero radius of convergence. They prove that the radius of convergence is greater than zero and give a lower bound for 3D Ising of 0.4, which corresponds to |beta J|=0.1, agreeing with what I gave in (8). This is to be compared with the accepted critical value of |beta J|=0.2216..., which is larger as it should be. I believe that [10] and my derivation are essentially the same when restricted to 3D Ising, although showing this will be a major enterprise.
Let us not go into [9] and [10] deeper, as that requires much more work than going through the derivation that I gave, which is in the same spirit, but directly for 3d Ising and in the more common Ising notations. Therefore, I do not claim that my proof is new.
My identity (4) follows from [11] or [12]. The clearest formulation is (6.1) in [12], which is proved in detail in the following equations. Suzuki gives a simple application in (6.5). Taking a more general choice for f and averaging over the spin g=sigma, you arrive at my (4). (Note that Suzuki uses S=sigma/2 for his spins.) This is the only identity that I need from the literature and its proof in [12] is easy to follow.
Comments on your numbered questions:
1) You repeat infinitely many times, but after p steps you have the first p series coefficients in beta exactly. After infinitely many steps you have the exact series in powers of a1, a3 and a5. And this series is absolutely convergent, while a1, a3 and a5 are also absolutely convergent series in t or beta.
2) You can replace "may" by "will", since it will happen infinitely many times. I just used more informal language. It connects with the following question.
3) As you replace one product of sigmas with new products of sigmas, it will happen that there are products where the number of sigmas for each site is even, so that the product is one. You stop with such terms as you have them exactly. You continue with the other terms.
4) In the infinite system you can not continue in one direction and return that way to where you started. In an N-by-N-by-N system the terms of order less than N can not depend on the system size, as you have not yet gone around the periodic boundary conditions. This is well-known and exploited by, e.g., Enting, Guttmann, and Jensen in their derivation of series expansions, with many results of them published in http://www.ms.unimelb.edu.au/~iwan/ .
5) There are infinitely many terms in the remainder. All these terms are small such that the sum of all their absolute values is bounded by the tail of a convergent geometric series (sum r^j), which goes exponentially fast to zero.
6) Searching for articles on exact upper bound of the critical temperature, I have found the following: J.O. Vigfusson, 18 (1985) 3417-3422; J.L. Monroe, J. Stat. Phys. 40 (1985) 249-257; L. Changfu, Phys. Rev. B 40 (1989) 11339-11341. They use various correlation function identities and inequalities, but do not address analyticity.
Best regards,
Jacques
我的回复(2009-4-7)：
Dear Jacques:
>Sorry for my later reply to your last e-mail, since I have been very busy for other things.
>Sorry for my later reply to your last e-mail, since I have been very busy for other things.
I have read carefully references [9] and [10].
> I cannot find the fine print about β = 0 in ref. [9]. Lebowitz and Penrose [9] indicated clearly in p. 102 of their paper that there is no general reason to expect a series expansion of p or n in powers of β to converge, since β = 0 lies at the boundary of the region E of (b, z) space. Their proof includes β = 0 only for hard-core potential (in section II of [9]), certainly not for the Ising model. Even in eq. (9) and also (10) of [9], they indicated clearly the condition of Re β > 0.
> Gallavotti et al. [10] proved that the radius of convergence is greater than zero, but once again their proof does not touch β = 0.
> Since you did not claim that your proof is new, may I understand that your proof is not better than these previous proofs? So, your proof for the radius of convergence still avoids the point of β = 0. It is also due to the fact I revealed in part II of my response that βf cannot be used to discuss the singularity at β = 0.
> I still have some questions on your comments 3:
> 1) How do you take into account the "internal factor" in your proof?
> 2) The system of eqs. (4)-(6) is viewed as a linear operator, how are these equations responsible for high-order "internal factor" terms?
> 3) When you cut off by ending the repeating process, in the meantime, do you cut off the non-local effect of the "internal factor"?
> 4) How do you use the condition of β = 0 in your proof? I hope to see the mathematical details, such as inequalities employed in the previous work [9,10].
> 5) One can easily read from your proof the information about the nearest neighbors, not dimensions. How do you distinguish the difference between simple cubic lattice and triangular lattice in your proof?
> Since you did not claim that your proof is new, may I understand that your proof is not better than these previous proofs? So, your proof for the radius of convergence still avoids the point of β = 0. It is also due to the fact I revealed in part II of my response that βf cannot be used to discuss the singularity at β = 0.
> I still have some questions on your comments 3:
> 1) How do you take into account the "internal factor" in your proof?
> 2) The system of eqs. (4)-(6) is viewed as a linear operator, how are these equations responsible for high-order "internal factor" terms?
> 3) When you cut off by ending the repeating process, in the meantime, do you cut off the non-local effect of the "internal factor"?
> 4) How do you use the condition of β = 0 in your proof? I hope to see the mathematical details, such as inequalities employed in the previous work [9,10].
> 5) One can easily read from your proof the information about the nearest neighbors, not dimensions. How do you distinguish the difference between simple cubic lattice and triangular lattice in your proof?
6) Furthermore, to point 5) in your last e-mail dated on April 1, 2009, infinitely many terms in the remainder go exponentially fast to zero. Is this exponential relation dependent of the distance between spins or the number of the terms?
7) According to point 6) in your last e-mail, the exact upper bound of the critical temperature, you found in those references, was not addressed analytically. It suggests that "This bound is a rough estimate and much better ones have been given in the literature" has not been supported by the references provided in your e-mail.
By the way, I found that results published in http://www.ms.unimelb.edu.au/~iwan/ concentrate only on planar lattices, not 3D ones. Would you please indicate clearly which reference is for our issue.
By the way, I found that results published in http://www.ms.unimelb.edu.au/~iwan/ concentrate only on planar lattices, not 3D ones. Would you please indicate clearly which reference is for our issue.
Best regards,
>
Zhidong
Perk教授的第10封来信(2009-4-7)：
Dear Zhidong,
You misunderstood us several times. About your blog: We are sad because we know that you still do not understand. We are sad for you, because it does/will hurt you. We are also sad because you are possibly misleading others with your errors. We hope that you will finally understand and not make more errors.
About your last email: First, following Yang and Lee, the Ising model is equivalent to a lattice gas with zero or one particle per site. This is a lattice version of a hard-core potential. So the statement of Lebowitz and Penrose does apply to Ising models, showing the analyticity about beta=0, if you make the correct translation. After that statement they cite Gallavotti et al. and consider themselves done with beta=0. Then they concentrate on showing analyticity and other properties in most of the phase diagram for beta not zero.
You do not seem to remember your calculus: If the radius of convergence in beta is greater than zero, then the function is analytic at zero and for any other complex number that has an absolute value smaller than the radius of convergence. As a simple example: f(x)=1/(1-x) has a radius of convergence equal to 1, so that 1+x+x^2+x^3... converges to f(x) as long as |x|<1. Your statement, that the proof does not touch beta=0 is wrong: beta=0 is within the radius of convergence and is, therefore, included.
When I say that my proof is not new, that does not mean that you can find it published in its form anywhere. What I mean is that the essential idea of the proof is the same as of other proofs, namely using linear correlation identities and bounding the resulting series absolutely with a geometric series. This then gives a lower bound for the radius of convergence, with beta=0 included at the origin of the circle in the complex plane. [You can do a similar proof for the low temperature series in powers of exp(-beta J), and find a lower bound for the radius of convergence of the spontaneous magnetization.]
About comment 3:
1) There are no internal factors to take account of if you work with the spins (sigma). They show up if you make a nonlocal Jordan-Wigner transformation going from spins to Gamma matrices. The internal factors are there only in the Gamma language, not in the sigma language.
2) System (4)-(6) only has a local effect. You only go to nearest neighbors of the spins in the correlation function you apply it to. Note that I work in the spin language!
3) You do not really cut off. After each step there are trivial correlations <1>=1. So you make an upper bound by assuming the two extreme possibilities that all terms are trivial and that no term is trivial. In the latter case you apply (4)-(6) again. You add upper bounds for the two extreme cases, which is a clear overestimate. This gives then a rough but rigorous upper bound in terms of an infinite geometric series.
4) Again, beta=0 is included as I derived a lower bound on the radius of convergence. Only what is larger is not included, namely the circle and the outside. Everything inside the circle, including its center beta=0, is addressed. This is basic calculus, first-year mathematics, that you still should remember. Even if I do not say beta=0 explicitly, the convergent power series is in powers of beta.
5) Here you are correct! The same proof applies to the triangular lattice and the simple cubic lattice.
6) In order n (after n steps) you have up to 32^n terms for each spin in each of the correlations in order n-1. But the total contribution of order is bounded by r^n + r^(n+1) + r^(n+2) + ... = r^n/(1-r). This includes the bound that all the terms in order n are trivial giving r^n, or none of the terms are trivial leading to r^(n+1) +..., coming from applying the linear identity (4)-(6) on each term. This is a clear overestimate. But all you need is this geometric series upper bound in order to show that all the infinitely many terms in all orders sum up to an absolutely convergent series.
It may be possible to show that no more than half of the terms in each order is trivial. That would lead to a lower bound on the radius of convergence that is twice as large. I did not try this. It is sufficient to show convergence in order to show that your statement of a phase transition at beta=0 is wrong.
7) It has not been rigorously established that there is a unique critical point, although all series, Monte Carlo and other methods strongly suggest that it is unique. The papers that I quoted prove that there is no spontaneous magnetization at higher temperature than their bound. Under the proviso that the critical point is unique, or at least coincides with the onset of spontaneous magnetization, much better bounds have been "given". Note that I did not say "proved". I said that they did not give a rigorous lower bound on the radius of convergence, as I said that they did not address analyticity.
Finally, there are several papers, e.g. H. Arisue and K. Tabata, "Low-Temperature Series for Ising Model by Finite-Lattice Method", Nucl. Phys. B (Proc. Suppl.) 42 (1995) 740-742 and references cited there that use the finite-lattice method to derive series for 3D Ising. The method was introduced by T. de Neef and I.G. Enting, [J. Phys. A 10 (1977) 801], for 1D and 2D models.
Because the linear identity is local, you can only move one step per order with nearest-neighbor interactions. That is why the finite-lattice method works in all dimensions. With periodic boundary conditions in all directions, you can only see the size dependence if you go around the periodic boundary conditions. With larger and larger lattices more and more terms are independent of the lattice size, whereas the remainder goes exponentially to zero [r^n/(1-r)]. Hence, there is no problem going to the thermodynamic limit and to prove analyticity for all beta with |beta| < lower bound on radius of convergence. Again, beta=0 is included!
Best regards,
Jacques
我的回复(2009-4-8)：
Dear Jacques:
First, I would like to thank your kindness. But please do not worry about me. I enjoy discussions with you and have learned lot of things from the discussions. I have my right to insist what I believe it is correct, in case that the objections are not prefect. It would not hurt me anymore. Also, please do not worry about others. I think that other scientists have their ability for making their own and correct judgments.
In Lebowitz and Penrose’s paper, they included beta=0 only in section II for hard-core potential, but never mentioned beta=0 for proof of the Ising model in other sections. I do not understand why they always stated the condition of beta>0 for the Ising model, if as you said, the hard-core potential model is completely equivalent the Ising model.
From all the references published, I learned that the condition for beta>0 is always employed for proof of the non-zero radius of convergence in beta, and if one applied beta=0, the key inequalities for such proof are invalided. This is why I do suspect such statement. In my opinion, the simple example: f(x)=1/(1-x), does not show the situation properly, since x = 0 can be simply used here for proof of a radius of convergence.
If the same proof applies to the triangular lattice and the simple cubic lattice, why the triangular lattice can be solved exactly, but not the simple cubic lattice? There should be some important factors being different in the two models, which should be taken into account in the proof.
As you stated, “It has not been rigorously established that there is a unique critical point, although all series, Monte Carlo and other methods strongly suggest that it is unique”. This would be one of the keys of all the troubles! A unique critical point has not been rigorously established yet! How can we believe the agreements between all approximation techniques (such as, series, Monte Carlo and other methods)?
Best regards,
Zhidong
Perk教授的第11封来信(2009-4-8)：
Dear Zhidong,
Lebowitz and Penrose want to increase the analyticity beyond a circle around beta=0. They do not say more about beta=0, as they did the continuous system in section II. They say that the lattice system goes the same way, but leave further details to Gallavotti et al., if the reader wants to check it out.
Again, you only need a nonzero lower bound on the radius of convergence. You do not have to look at beta=0. The formulas in the literature have well defined limits for beta to zero, that are not zero. You may get 1/0 or infinity. Then you conclude that the radius of convergence is greater or equal to zero. There is no problem, only that this bound is not good enough and you have a better one with |beta|>0. By the way, in the proof in my version 4 there is no problem with the formulas for beta=0, so that your concern does not even come up there.
That the proof applies to both cubic and triangular is because of the local nature of the spin language. The obtained lower bound on the radius of convergence is far from optimal. Once you would go make more sophisticated estimates, the difference of the two lattices will show up. The same is true between the square lattice and the diamond lattice, which both have coordination number 4. Both high-temperature series start the same, but in higher order the trivial <1> correlations occur in different terms, so very soon at very low order the series already become different.
Finally, all approximate methods done to equivalent high order of approximation are consistent with each other, and with the existence of a unique Tc above which f or beta*f is analytic. (Again, at T=infinity you must look at beta*f, as the kt=1/beta factor in f is only an artificial complication.) Also your wrong 3D result gives only one Tc>0. If this point is a worry to you, then your 3D result already fails here.
Best regards,
Jacques
我的回复(2009-4-8)：
Dear Jacques
It is hard for me to judge from the proof in your version 4 whether there is no problem with the formulas for beta=0, since you did not provide mathematical details for your proof, specially, about how it works at beta=0.
However, if one started from |beta|>0, then the series expansion of my conjectured solution would be supported also by the proof of the radius of convergence.
In such high-temperature series, how do the dimensions act? How are the non-local topologic problems dealt with?
In my last e-mail, I misunderstood the meaning of “a unique critical point”.
Best regards,
Zhidong
Perk教授的第12封来信(2009-4-8)：
Dear Zhidong,
I do not understand that there is not enough detail. It is easy to fill in the details: Eq. (4) follows from Suzuki and you should have understood it as it is not hard. Eqs. (5)-(7) are easy to check, especially if you use computer algebra. The alternating sign property is also easily checked, expanding (7) as sum of three terms a_j/(1+b_j t^2) for j=1,2,3. Again, if you use computer algebra it is trivial, but even by hand it is not too bad. Then you see that the largest absolute value of (7) follows if t^2 is negative. Next put t=ir in (7) and solve for r when(7) is i=sqrt(-1). The smallest answer is given in (8), giving the desired lower bound of the radius of convergence. You get for any nontrivial <...> that |<...>| < r max(|<...>_next|), which is also < r (1 + max(|<...>_next-nontrivial|) ). Here next means the 32 correlations in the RHS of (4)-(6). Continuing you get < r +r^2 (1 + max(|<...>_nextnext-nontrivial|) ). Continuing you get r + r^2 + r^3 +..., a convergent geometric series showing that the series in t=tanh(beta J) is absolutely convergent, as long as (8) is obeyed. Also beta=0, which is t=0, obeys (8). You get 0+0+0+...=0, which is a triviality. The full series is sum of a_j beta^j jor j=1,2,...,infinity, and sum |a_j beta^j| converges, if (8) holds. More on the rest is in my previous emails. Because of the analyticity, your solution should have the same first few series coefficients as derived in the literature, and my version (4) gives an alternative way to find them. Dimension and topology of the lattice act through the values of the a_j's that depend on them, apart from the first very few. But my lower bound r on the radius of convergence giving a circle in the complex beta plane around beta=0 where beta f is analytic is independent of dimension and topology. It depends only on the coordination number (number of nearest neighbors).
Best regards,
Jacques
PS: You should have enough information. Just take time to think it over. I cannot reply further in the next several days.
我的回复(2009-4-8)：
Dear Jacques
Thanks again for your detail reply about your proof. I shall think it over
Thanks again for your detail reply about your proof. I shall think it over
I still have few questions.
1) In the last paragraph of section II of Lebowitz and Penrose' paper, they stated like this, "For lattice systems, a proof implying such convergence ... [24] and analyticity for ... is proved in [12]." Why did they say "implying " if such proof in [24] is rigorous?
However, as I pointed out already, in [12], Gallavotti et al did not use beta = 0 for their analyticity.
1) In the last paragraph of section II of Lebowitz and Penrose' paper, they stated like this, "For lattice systems, a proof implying such convergence ... [24] and analyticity for ... is proved in [12]." Why did they say "implying " if such proof in [24] is rigorous?
However, as I pointed out already, in [12], Gallavotti et al did not use beta = 0 for their analyticity.
2) For "It has not been rigorously established that there is a unique critical point, " would you find a reference for this statement? If a unique critical point were not established, would it be possible that there may exist two phase transitions (not two Tc)?
Best regards,
>
Zhidong
>
Zhidong
Perk教授的第13封来信(2009-4-12)：
Dear Zhidong,
About your two questions:
Question 1: First about reference [24] with small |z|: Gallavotti and Miracle-Sole work in the lattice gas language, and they have |z| small, or H large, z=exp(-2 beta H). They do the fugacity expansion (z-expansion), also known as the high-magnetic field expansion. This is also the Mayer expansion for which Groeneveld gave the first convergence proof as part of his thesis ……
Their proof "implies": They get absolute convergence, and you can give further details from their estimates for the beta expansion, for H real and sufficiently large (z small). To get to all real magnetic field H, especially H=0 (z=1), you need the rigorous estimates given in [12]. There they address the beta expansion.
You want the radius of convergence positive, so that beta=0 is well within the convergence circle in the complex beta plane. Therefore, [12] does not need to mention beta=0 explicitly. Analyticity is proved for all complex beta with |beta| < radius.
Question 2: I gave you my references already: Proofs of analyticity up to some beta >0 (see ref. [12] and my addition to the rejoinder, e.g.) and also proofs that there is no spontaneous magnetization for beta's up to a much larger value. This leaves gaps between the theorems and the numerical studies of 3D Ising. The consensus is that there is only one Tc above which there is analyticity and below which there is spontaneous magnetization. But this is not rigorously proved, only supported by many numerical studies. To give a proof would be very close to giving an exact solution.
Best regards,
Jacques
……………………………………………….
我的回复(2009-4-12)：
Dear Jacques:
Thanks for your message and also for your kindness!
Best wishes, also to Helen,
Zhidong
学术论剑
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