我于2024年12月8日将三维Z₂格点规范场理论精确解的论文投稿到arXiv。经过arXiv审核员的审查，于2024年12月17日公开发表（arXiv: 2412.10412）。当天就收到Slava Rychkov的一封来信：

Dear colleague

I noticed your paper today. I was wondering what is the relation of your work to the very accurate Monte Carlo simulations by Hasenbusch and the conformal bootstrap results for the 3D Ising critical exponents, which do show that the exponents do not take the values you propose. The most accurate value of eta is not at all 1/8 but 0.036297612(48) (which by the way can be neatly expressed in terms of some gamma functions, see https://en.wikipedia.org/wiki/Ising_critical_exponents)

It would be good if you at least cited that work with which you an in manifest disagreement and explain what in your opinion is wrong with those approaches.

Best Slava Rychkov

    这个Slava Rychkov是何许人也？斯拉娃·雷奇科夫是法国高等科学研究所的物理学家。主要研究领域是理论物理学，致力于发展强耦合量子和共形场论的研究方法。主要工作是发明了一个所谓的“共形靴带” (conformal bootstrap)方法研究伊辛模型以及其他物理模型中的临界现象，号称得到“最精确的”临界指数。雷奇科夫还研究随机场伊辛模型和帕里西-索利斯超对称理论。斯拉娃·雷奇科夫是伊辛模型临界指数研究方面一颗非常耀眼的“学术明星”。在Phys.Rev.Lett., Phys. Rev.等国际学术刊物发表论文140篇，其中有三篇论文引用超过500次。2019年1月11日、2024年11月21日在Rev. Mod. Phys.发表两篇共形拔靴数值模拟的综述论文，风头正劲。在2012年雷奇科夫的第一篇共形拔靴论文发表后，我就关注到这个工作。我理解，所谓的“共形靴带”就是利用伊辛模型等在临界温度附近的共形不变性研究临界现象的标度不变性，整体的理论框架仍然在重整化群理论模拟范围内，并没有考虑到三维伊辛模型中非平庸拓扑结构的贡献。我当时就有一个预感，与共形靴带这伙人早晚有一战。现在，该来的终于来了。

在这个email中雷奇科夫认为我的临界指数与Hasenbusch的非常精确蒙特卡洛模拟以及共形靴带的结果不一致。明里暗里要求我引用他们的工作，并且要说明为什么与他们的工作不同。这一篇email来势汹汹，挑衅意味十足，并且提出不合理的引用要求。

2024年12月18日我的回信：

Dear Dr. Slava Rychkov,

Many thanks for your kind interest on my paper.

Since the publication of my original paper on two conjectures, I have been always asked to compare my solution with other methods, such as renormalization group, Monte Carlo, series expansions, etc. Indeed, I have already compared with these approximations in my previous papers, for instance, on pages 5374-5389 in [Philosophical Magazine, 87 (2007) 5309–5419], on pages 3-5 in [Journal of Physics: Conf. Series 827 (2017) 012001]. A common question is that why all these methods, which are consistent with other others, are incorrect, but my solution is correct. Now, I can answer the question as follows: all the approximations neglect the same effect: the nontrivial topological effect in the many-body spin interaction systems.

For the conformal bootstrap, I understand that it has the highest precision among all the approximation methods, but the same as others, with a systematical error. The error originates from missing the contribution of global effect to physical properties, as revealed in my work. This is a fundamental contribution that has never been taken into account in all the approximation methods including the conformal bootstrap.

As attached a paper of Prof. Wei Wang, by use of Monte Carlo method, following my idea, the simulations give the critical exponents the same as the exact solution. Furthermore, I collected some experimental data in literatures (most published in recent 10 years), which agree well with my solution. Please find the attached figure for such a comparison.

Any further comments/discussion from you are greatly appreciated.

Best regards,

Zhidong Zhang

在这一封回信中我表达了以下观点：自从我的猜想论文发表后，我总是被要求将我的解与其他方法（如重整化群理论、蒙特卡洛、级数展开等）做比较。我在前期论文中（指出具体的页码范围）已经做了比较。一个经常性的问题：为什么结果一致的这些方法是错误的，而精确解是正确的？答案是：所有的近似方法忽略了相同的效应：在多体自旋相互作用体系的非平庸拓扑结构。共形靴带方法在这些近似方法中具有最高的精度，但是与其他方法一样具有一个系统误差。误差来源于忽视全局效应对物理性质的贡献。这是一个基础性的贡献，在以前的所有近似方法（包括共形靴带）均没有计及这个贡献。我通过附件发给他王威教授的蒙特卡洛模拟结果、精确解与实验的比较，表明我的结果与王威的结果以及实验一致（相关内容见博文《终结猜想-33-量子相变》）。

很快Slava Rychkov当天回信：

Dear Dr Zhidong Zhang,

I understand you may have doubts about conformal field theory, since it's not a first principle technique but requires nontrivial theoretical input.

But what about MC simulations or high-T expansion of exactly the model that you purport to have solved - the Nearest Neighbor 3D Ising model on the cubic lattice? There are hundreds of such studies, which are by and large consistent among themselves and not consistent with your purported solution.

Wang's simulations do not consider the 3D Ising model, it seems. It appears that he simulated another model with an extra long-range interaction not present in the original model. When we include long-range interactions of course critical exponents may change. Shouldn't you compare to MC simulations of the model that you claim to have solved, not to some other model?

In experimental systems, sometimes long-range effect are present. So it's not surprising to me that some experiments may agree with your exponents.

Best Slava

Slava Rychkov这一封来信的大意：承认共形场论不是一个第一原理的技术，要求非平庸的理论输入。但是蒙特卡洛和高温展开研究的体系与我的模型一样，有数百个研究的结果相互一致，而与我建议的解不一致。王威的模型已经不是三维伊辛模型，加了长程相互作用就不是原来的模型了。实验结果不说明问题，有时候会出现长程相互作用，从而与我的解一致。

    这里她不打自招，承认了她的理论的弱点！需要非平庸的理论输入，也就是要通过调节参数进行拟合，那当然能得到很高的精度了。但是，高精度不等于高精确度。存在系统误差是致命的错误。她的理论也就是一个唯象理论的水准。她认为王威改变了模型，这是她不理解问题的本质。三维伊辛模型存在非平庸的拓扑结构、长程自旋纠缠，等效于长程自旋相互作用，只有加上这一项拓扑学的贡献才能真正表示三维伊辛模型。我随即做了如下的答复：

Dear Slava,

As revealed in my work, although the 3D Ising model has the nearest neighbor interactions only, there exist long-range spin entanglements (namely, nontrivial topological structures), please refer to my Z₂ lattice gauge paper for the origin of this effect. MC simulations or high-T expansion treat only the local effect of the model. Due to the size limit, MC simulations cannot take into account the nonlocal effect of the 3D Ising model. In my previous papers published in Mathematics with Prof. Suzuki, we indicated that the long-range entanglements are effectively equivalent to the spin chains along the third direction. Prof. Wei Wang follows this ideal to simulate the model with an extra long-range interaction. Indeed, as what Suzuki and I predicted, such extra long-range interaction effectively equals to the effect of the long-range entanglements.

Once again, I have to answer the question: why are hundreds of studies consistent among themselves and not consistent with my purported solution. My answer is still the same: all of them do not consider the contribution of the nontrivial topological structure to physical properties of the 3D Ising model with the nearest neighbor interactions.

Everyone thinks that 1 + 1 = 2, and believes it is correct. But, the solution of the problem is that 1 + 1 + 1 = 3. 

Best,

Zhidong

在我的第二封回信中表达了以下观点：尽管三维伊辛模型仅仅包含最近邻相互作用，其中存在长程自旋纠缠（非平庸的拓扑结构），请看我的Z₂格点规范理论论文中对这个效应起源的描述。蒙特卡洛模拟或者高温展开仅仅考虑模型的局域效应。由于尺寸的限制，蒙特卡洛模拟无法计及三维伊辛模型中的非局域效应。在我与铃木理教授合作的论文中，我们指出长程自旋纠缠有效地等价于沿着第三维度方向上的自旋链。王威教授按照我们的思路模拟了具有附加长程相互作用的模型，获得我们预言的结果：这样的长程相互作用有效地等价于长程纠缠效应。每一个人认为1 + 1 = 2，并且相信是正确的，但是问题的解是1 + 1 + 1 = 3。 

2024年12月18日Slava Rychkov再次答复：

Dear Zhidong,

this does not agree with my understanding of Monte Carlo simulations, which are unbiased methods probing all effects there are in the system. If you believe the MC simulation is wrong, or that it misses some effect (for example some configurations are not probed), you should modify a Monte Carlo algorithm, find a better algorithm, not change the model. But I believe it's mathematically proven that Metropolis algorithm is a good algorithm. So I don't see much room for your scenario to be true.

But thanks for the exchange.

Best regards, Slava

    Slava Rychkov这一封来信的大意：我的答复与她理解的蒙特卡洛模拟不相符。蒙特卡洛模拟可以公正地探测体系的所有效应。如果我相信蒙特卡洛模拟是错误的，或者它丢掉了一些效应（例如没有探测到一些组态），我应该改进蒙特卡洛的算法，找到一个更佳的算法，而不是改变模型。但是，她相信蒙特卡洛模拟使用的Metropolis算法数学上被证明是一个好算法。所以，她不认为我的结果是正确的。

    在这里，Slava Rychkov仍然认为我们改变了模型。这是她认识上的误区。他们的方法没有反映三维伊辛模型的拓扑学贡献，他们才是改变了模型。尽管他们的研究是从一个原始的三维伊辛模型出发，但是无法体现模型的内禀性质（长程自旋纠缠、非平庸拓扑结构）。另外，把蒙特卡洛模拟使用的Metropolis算法拽出来当挡箭牌。并且向我提出一个挑战性的任务：找到一个比Metropolis算法更佳的算法。

2024年12月19日我的回复：

Dear Slava,

For Monte Carlo simulations as well as Metropolis algorithm, please see the detailed discussion on pages 5385-5387. For any algorithm, one must face the problem of exponentially growing computational complexity with the size N of the system. In a 3D Ising model, the total states are 2^N, so the computational complexity would be in the same order. Any algorithm cannot deal with such a problem as the size N approaches infinite. Monte Carlo simulations usually can simulate only a cell with finite size, while taking the periodic boundary conditions assuming all the finite cells have the same behaviors. Clearly, this procedure breaks the long-range spin entanglements (namely, nontrivial topological structures), and certainly misses the global information of the many-body spin interacting system.

It is not my job to modify a Monte Carlo algorithm, or to find a better algorithm. My job is to understand well the physics in the system. Actually, my understanding on the effect of the nontrivial topological structures helps me to solve an important problem in mathematics and computer science. Attached please find two papers that solve the NP-complete problems. Again, any comments/discussion from you are greatly appreciated.

Best regards, Zhidong

在我的第三封回信中表达了以下观点：有关蒙特卡洛模拟以及Metropolis算法，请见我的猜想论文（给出具体页码）。任何算法都必须面对随着体系尺寸N的增加指数增长的计算复杂度问题。三维伊辛模型的总状态数为2^N，计算复杂度具有相同量级。当N趋近于无限大时任何算法都无法处理这样的问题。蒙特卡洛模拟通常仅仅能够模拟有限尺寸的晶胞，同时应用周期性边界条件，并且假定所有的晶胞具有相同的性质。很显然，这个过程打断了长程自旋纠缠（非平庸拓扑结构），肯定失去了多体自旋相互作用体系的全局效应。我的工作不是改进蒙特卡洛算法，或者找到一个更好的算法。我的任务是理解好体系中的物理。实际上，我有关非平庸拓扑结构效应的理解帮助我解决了数学和计算机领域一个重要问题。附件发给他我的有关NP完全问题的两篇论文，欢迎评论和讨论。

    然后，然后就没有然后了。

通过这一次的学术交锋，大呆有几点收获：1）做科学研究还是要踏踏实实追求真理，靠炒作、吹泡泡不能长久，经不住推敲，热闹过后早晚凉凉。不要被一些热点和繁荣所迷惑，不要过分迷信高档刊物、高引用。高档刊物（包括CNS, Phys.Rev.Lett.、Rev.Mod.Phys.等）上有垃圾。2）这一次的学术交锋可以称为“闪击战”，几个回合解决战斗。刺破了“学术明星”的光环，挤掉了“学术泡泡”。最主要原因是，我站在真理一边。另外，经过以前与四大天王、Perk等的交锋，大呆的功力大涨（见《激辩猜想-1-开篇》系列博文），具备了一招制敌的能力。3）不指望这些从事共形拔靴模拟的科学家改变态度，他们在学术界具有一股强大的势力，自以为取得了很大的进展，正是当红。自娱自乐自嗨在错误的道路高歌猛进。不仅仅在伊辛模型研究领域存在这样的现象，在其他领域也同样存在。一些学术明星创造出一些偷梁换柱的“新概念”，在高档刊物发表，人为制造出新的热点，就会引发一群人跟风。一想到许多人没有独立的思维，盲目跟风而浪费了生命，大呆不禁心生怜悯。

当然，这一次学术交锋仅仅给《终结猜想》系列博文带来了一个小插曲。在具体地址求解了二维横场伊辛模型精确解（见《终结猜想-33-量子相变》）、三维Z2格点规范理论精确解（见《终结猜想-34-三维Z2格点规范理论》）之后，一个重要的步骤是建立一个普适性的理论，将求解三维伊辛模型精确解过程中获得的物理思想和数学结构融入到新的理论中去。大呆的下一个目标是：拓扑量子统计物理学。

请见下回《终结猜想-36-拓扑量子统计物理学》。

相关论文：

          1,提出两个猜想：Z.D. Zhang, Philosophical Magazine 87 (2007) 5309. https://doi.org/10.1080/14786430701646325

       2,初探数学结构：Z.D. Zhang, Chinese Physics B 22 (2013) 030513.

https://doi.org/10.1088/1674-1056/22/3/030513

3,证明两个猜想-克利福德代数方法：Z.D. Zhang, O. Suzuki and N.H. March, Advances in Applied Clifford Algebras 29 (2019) 12. https://doi.org/10.1007/s00006-018-0923-2

4,证明猜想1-黎曼-希尔伯特问题方法：O. Suzuki and Z.D. Zhang, Mathematics, 9 (2021) 776. https://doi.org/10.3390/math9070776

5,证明猜想2-黎曼-希尔伯特问题方法：Z.D. Zhang and O. Suzuki, Mathematics, 9 (2021) 2936. https://doi.org/10.3390/math9222936

6,自旋玻璃三维伊辛模型计算复杂度： Z.D. Zhang, J. Mater. Sci. Tech. 44 (2020) 116.  https://doi.org/10.1016/j.jmst.2019.12.009 

7,二维横场伊辛模型的精确解：Z.D. Zhang, Physica E 128 (2021) 114632. https://doi.org/10.1016/j.physe.2021.114632

8,拓扑量子统计物理和拓扑量子场论： Z.D. Zhang, Symmetry, 14 (2022) 323.

https://doi.org/10.3390/sym14020323

       9,布尔可满足性问题计算复杂度，Z.D. Zhang, Mathematics, 11 (2023) 237. https://doi.org/10.3390/math11010237 

     10. 黎曼z函数与伊辛模型零点分布的等价性：Z.D. Zhang, arXiv:2411.16777.

https://arxiv.org/abs/2411.16777

11. 三维Z₂格点规范场理论精确解：Z.D. Zhang,arXiv: 2412.10412.

https://arxiv.org/abs/2412.10412