# A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{B}T) > 0, not for β = 0. The so-called exact low-temperature expansion is evidently divergent, which indicates that this expansion approach is not exact, not only for its high-order terms, but also for the first term, as the approach itself is questionable. The Lee–Yang Theorem for phase transitions offers a possibility of a phase transition at infinite temperature in the Ising models [28,29], which provides a possibility of multi-valued functions for high-temperature expansions. Note that the convergences of βf and f are different at/near infinite temperature. The approximation methods (including perturbative, nonperturbative) and computer simulations (including Monte Carlo) do not take into account the contribution of the non-trivial topological structure of the 3D Ising model to the physical properties. Missing the global effect is the main reason that all these approximation methods consist with each other, but are incorrect due to the existence of systematical errors, no matter how high precision they achieve. The systematical errors of these approximation techniques are related directly to the physical conceptions/pictures at the first beginning and the neglects of important non-locality factors during procedures. As pointed out in [30] that these estimates in [17] were obtained based on certain hypotheses (e.g., the existence of a sharp kink) and that if these hypotheses are not used, then the conformal bootstrap analysis appears to be consistent with the values η = 1/8 and ν = 2/3, obtained by Grouping of Feynman Diagrams, which are consistent with the Zhang’s solutions obtained in [3]. Furthermore, Zhang’s results agree with some experimental results, which are carefully performed with high accuracy (see in [31], for instance). After its publication [3], Zhang’s conjectured solution has received supports from several groups, for instance, March and his co-workers [32,33,34,35,36], Ławrynowicz and some mathematicians [6,7,37,38], Kaupuzs and his colleagues [30,39,40], and others [41,42,43,44,45,46,47,48,49,50,51,52,53]. In [8], Zhang-Suzuki-March rigorously proved four Theorems, which verifies the correctness of the Zhang’s conjectured solution [3]. The correct way to judge the correctness of an exact solution is to check whether there is anything wrong in the deriving process of proofs of Theorems, not to judge it by the approximant results. After the publication of Zhang-Suzuki-March’s work [8], up to date, no further criticisms have been published. It was suggested that the approximation techniques can be utilized to obtain the non-locality part of the partition function (as well as the thermodynamic physical properties) by extracting the approximation values from the exact solution [54,55].

## 2. 3D Ising Model and Zhang’s Conjectures

#### 2.1. Hamiltonian and Partition Function of 3D Ising Model

_{1}+ H

_{2}+ H

_{3}

_{3}) appear in the partition function, which represent the existence of a long-range many-body entanglement between spins in the ferromagnetic 3D Ising model, because of the nature of three dimensions.

#### 2.2. Zhang’s Conjecture 1

^{N}states of a diagram with N crossings [57,58]. The bracket state summation is an analog of a partition function in discrete statistical mechanics, which can be used to express the partition function for the Potts model for appropriate choices of commuting algebraic variables [57,58]. This means that not only the local spin alignments but also the crossings of knots contribute to the partition function Z of the 3D Ising model in the zero external field. The contribution to the partition function Z by knots also reflects the entropy cost of tying knots, as the partition function Z is related with the free energy F, the internal energy U, and the entropy S.

_{3}and take a knot γ which is constructed by the horizontal line and vertical joint line with vertex {P

_{j}}, which is denoted by γ = {P

_{j}}, as illustrated below in Figure 1, for example.

**V**,

_{1}**V**, and

_{2}**V**are associated to some points in γ, which are denoted by X = {V

_{3}_{i}

^{(j)}}. Then, we have a data (γ, X) which is called Knot/Clifford (K/C) data, ${Z}_{\gamma}$. The Zhang’s conjecture 1 can be stated in the following manner:

- For a given ${Z}_{\gamma}$, can we make a trivialization ${\tilde{Z}}_{\tilde{\gamma}}$?
- More exactly, we can state the Zhang’s conjecture 1 as:

#### 2.3. Steps to Prove Zhang’s Conjecture 1 on Trivialization

- (1)
- The Clifford algebra $Cl({I}_{3D})$ is extended to the K/C algebra which has the original Clifford algebra and its conjugate algebra $\overline{Cl({I}_{3D})}$ as subalgebras (Section 3).
- (2)
- ${Z}_{\gamma}$ is extended to the K/C algebra which is denoted by $\sigma ({Z}_{\gamma},{\overline{Z}}_{\gamma})$. Therefore, we have a knot carrying the elements in K/C algebra for the partition function (Section 4).
- (3)
- (4)
- Applying the monoidal transformation to the solution in (3), we construct the desired trivialization in K/C algebra (Section 7).

## 3. Clifford Algebra of the Ferromagnetic 3D Ising Model and Its K/C Algebra

#### 3.1. Clifford Algebra of the Ferromagnetic 3D Ising Model

_{j}(j = 1,2,…, 2N−1):

_{j}(j = 1,2,…,2N+1).

_{2}), ${s}^{\prime}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ (= σ

_{3}), $C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ (= σ

_{1}). Following the generation process of Clifford algebra based on C, we obtain the generation

#### 3.2. K/C Algebra Associated to the Knot Structure

**Proposition:**

- (1)
- Knot/Clifford algebra is an associative algebra.
- (2)
- We have${\mathsf{\Gamma}}_{i}{\mathsf{\Gamma}}_{j}={\mathsf{\Gamma}}_{i}{\overline{\mathsf{\Gamma}}}_{j}$,${\overline{\mathsf{\Gamma}}}_{i}{\mathsf{\Gamma}}_{j}={\overline{\mathsf{\Gamma}}}_{i}{\overline{\mathsf{\Gamma}}}_{j}$. Here, we have calculated the product including${s}^{\prime},{s}^{\u2033}$as the usual matrix calculus. It implies that the first element determines the sequence to be Γ-sequence or$\overline{\mathsf{\Gamma}}$–sequence.

## 4. Knot Structure of the Ferromagnetic 3D Ising Model

#### 4.1. Knots with Clifford Algebra Data

_{1}, …, P

_{n}}. Namely, γ = P

_{1}, …, P

_{n}. Choosing elements ${X}_{1},\dots {X}_{n},{X}^{\prime}{}_{1},\dots {X}^{\prime}{}_{n}\in Cl({I}_{3D})$, and making their conjugates ${\overline{X}}^{\prime}{}_{1},\dots {\overline{X}}^{\prime}{}_{n}$. (${X}_{i}$,${\overline{X}}^{\prime}{}_{i}$) is associated to γ at P

_{i}. The sequence is denoted by

#### 4.2. Association of Circle/Interval to Γ-Factors

#### 4.3. Generation of Knots of Ferromagnetic 3D Ising Model for Partition Functions

**V**and

_{1}**V**are omitted for simplicity. The elements ${i}^{k}({\mathsf{\Gamma}}_{i1}{\mathsf{\Gamma}}_{j1})\dots \dots ({\mathsf{\Gamma}}_{ik}{\mathsf{\Gamma}}_{jk})$ (and ${i}^{k+1}{\mathsf{\Gamma}}_{i}({\mathsf{\Gamma}}_{i1}{\mathsf{\Gamma}}_{j1}\dots \dots {\mathsf{\Gamma}}_{ik}{\mathsf{\Gamma}}_{jk}){\mathsf{\Gamma}}_{j}^{}$) (with j = 1,2,…,nl; k = 2n) together with their conjugate elements are called basic form of type II (see Figure 3), which can be represented as a braid with many crosses (k = 2n).

_{2}**V**is equal to the Pauli matrix $-{\sigma}_{k}^{z}$ [13], contributing a crossing topologically [57,58]. Therefore, each term of exponential elements in the transfer matrices

_{3}**V**and

_{1}**V**contributes a circle to the knot structure, while each term of exponential elements in

_{2}**V**contributes a braid. There is also a type which is defined by the product of exponential elements generated by the basic types above. The circles in

_{3}**V**and

_{1}**V**can be adjoined together with the lattice points of the normal knots γ, while the braids in

_{2}**V**can be connected as the product type of knots. As an example, the following figure (Figure 4) just shows three of the braids in

_{3}**V**connecting to the lattice points of the knot γ, while it does not show the circles of

_{3}**V**and

_{1}**V**for simplicity.

_{2}**V**forms new crosses, making the topological structure much more complicated. We can introduce a concept of confluency of knots and discuss the generation of K/C knots by use of successive confluence operations, to obtain the K/C knots of the generators

_{3}**V**(i = 1,2,3) of the partition functions of the ferromagnetic 3D Ising model in the zero external field. Nevertheless, the process for trivialization of the above topological structure can be employed directly to trivialize the more complicated one, as the concept, the principle, the role, and the process are kept the same.

_{i}**Theorem**

**I.**

**V**and

_{1}**V**contribute the trivial parts to the topological structure of the ferromagnetic 3D Ising model in the zero external field, while the transfer matrices

_{2}**V**contribute the non-trivial knots to the system.

_{3}## 5. Realization of Knots on a Riemann Surface

#### 5.1. Realization of Knots

_{1}, a

_{2}, ……, a

_{n}. We prepare two copies of complex projective space P

^{1}. A 2-covering Riemann surface M

_{g}is made by use of cut-segments:

#### 5.2. Realization of Knots on a Four-dimensional Manifold

_{g}in P × C as a covering space over P

^{1}(Figure 8). Then, we assume that the knot has a singularity of normal crossing.

#### 5.3. Realization on a Riemann Surface

_{j}: ${\mathrm{C}}_{\epsilon}^{(j)}=\left\{\left|z-{a}_{j}\right|=\epsilon \right\}$. Assume that ${C}_{\epsilon}^{(j)}\cap {C}_{\epsilon}^{(k)}=\varphi (j\ne k)$

- (1)
- For each a
_{j}, we take ${\tilde{a}}_{j}^{+},{\tilde{a}}_{j}^{-}$ on ${\mathrm{C}}_{\epsilon}^{\left(\mathrm{j}\right)}$ corresponding ${a}_{j}^{+},{a}_{j}^{-}$, respectively. - (2)
- Starting from ${a}_{1}^{+}$, we take ${\tilde{a}}_{1}^{+}$ on the upper surface.
- (3)
- The next element is denoted by α
_{k.}When ${\alpha}_{k}={a}_{k}^{+}$, we take ${\tilde{a}}_{k}^{+}$ and joint them without cut segment (Figure 9).

- (4)
- Repeating this process, we obtain a closed curve which is located very near to the original knot curve.

**Remark**

**1.**

**Example.**

^{1}and make cuts between $\overline{{a}_{1}{a}_{2}}$, $\overline{{a}_{3}{a}_{4}}$, and make a hyper-elliptic curve. Making small circles at ${a}_{1},{a}_{2},{a}_{3},{a}_{4}$, and following the generation scheme in Figure 12, we can obtain the realization of the knot in Figure 11 on the Riemann surface.

**Proposition.**

## 6. Method of Riemann–Hilbert Problem for 3D Ising Model

#### 6.1. Riemann-Hilbert Problem

_{g}which has a regular singularity at a

_{1}, a

_{2}, …, a

_{M}. The function has the following form:

**Theorem**

**(H.**

**Röhrl**

**[59]).**

_{j}(j = 1,2,……,M) which realizes the given representation, i.e.,${\gamma}^{*}f(z)=\rho (\gamma )f(z)$for any closed path γ.

**Remark**

**2.**

#### 6.2. Riemann–Hilbert Problem for the Ferromagnetic 3D Ising Model

**Theorem**

**II.**

**V**(i = 1,2,3) of the partition function of the ferromagnetic 3D Ising model in the zero external field, we find a knot${\gamma}_{i}$, which is given in Section 4. After making the realization of knots on a Riemann surface, we can find the following representation:

_{i}**V**(i = 1,2,3) by${\gamma}_{i}$.

_{i}**V**.

_{i}**Remark**

**3.**

## 7. Construction of Trivialization by Monoidal Transforms

#### 7.1. Direction Separation—Basic Idea

^{2}→ R

^{2}.

#### 7.2. Monoidal Transform

_{0}, we choose a local coordinate ${U}_{\epsilon}\left({P}_{0}\right)\left(=\left\{\left(x,y\right)|{x}^{2}+{y}^{2}<\epsilon \right\}\right)$. A complex manifold ${\widehat{U}}_{\epsilon}({P}_{0})$ is made to satisfying the following condition: There exists a mapping (see Figure 14)

- (i)
- Q
^{−1}(P_{0}) ≅ Ρ^{1}(complex projective space) - (ii)
- ${Q}_{{P}_{0}}:{\widehat{U}}_{\epsilon}\left({P}_{0}\right)-P\cong {U}_{\epsilon}\left({P}_{0}\right)-\left\{{P}_{0}\right\}$

#### 7.3. Complex Line Bundle of Monoidal Transformation

_{0},V

_{∞}} be the standard local coordinate system of Ρ

^{1}: $P={V}_{0}{\displaystyle \cup}{V}_{\infty}$

#### 7.4. Construction of Monoidal Transform

**Remark**

**4.**

^{2}(= R

^{4}), the manifold$\tilde{M}$is not Euclidean and non-trivial topology appears.

#### 7.5. Basic Notations on Trivialization

- (1)
- Trivial elements of exponential type

- (2)
- K/C mapping

#### 7.6. Basic Idea on Trivialization

_{1}in Figure 15:

**Theorem**

**III.**

**V**=

**V**The process can be performed in a completely analogous manner, but it is much more complicated, as there are many products of knots.

_{1}V_{2}V_{3.}**Remark**

**5.**

^{nl}possible solutions in 2

^{nl}sub-spaces produced by the direct product of all the sub-transfer matrices. The combination of the local transformation and the Largest Eigenvalue Principle solves the problems of overdetermined tetrahedron equations. The exact solution we found is $K{K}^{*}=K{K}^{\prime}+{K}^{\prime}{K}^{\u2033}+{K}^{\u2033}K$ [3], which is a star–triangle relationship and a solution of the tetrahedron equations and also the Yang–Baxter equations [4,8].

## 8. Construction of Solution to the Zhang’s Conjecture 1

- (1)
- We assume that a knot γ is given on the lattice: γ = {P
_{i}}, where P_{i}is on the lattice points. - (2)
- We take a partition function which is defined by the transfer matrices
**V**,_{1}**V**, and_{2}**V**The members are denoted by V_{3}.^{(i)}(i = 1,2,…, M). - (3)
- We distribute V
^{(i)}(i = 1,2,…, M) on knot points P_{i}of γ, and then have K/C knots Z_{γ}= (γ, X), with X = {V^{(i)}}. - (4)
- Making the K/C algebra, we make the conjugate element ${\overline{Z}}_{\overline{\gamma}}=(\overline{\gamma},\overline{X})$ and introduce a K/C knot $\sigma ({Z}_{\gamma},{\overline{Z}}_{\overline{\gamma}})$
- (5)
- After realization of the knot $\sigma ({Z}_{\gamma},{\overline{Z}}_{\overline{\gamma}})$ on a Riemann surface, we can formulate the Riemann–Hilbert problem for the representation.
- (6)
- We find the solution to the problem with regular singularities at the knot points of the realized knot on the Riemann surface which is denoted by $[{Z}_{\gamma}^{reg},{\overline{Z}}_{\overline{\gamma}}^{reg}]$.
- (7)
- Applying the monoidal transforms at the knot points, we obtain the trivialization $[{\tilde{Z}}_{\gamma}^{reg},{\tilde{\overline{Z}}}_{\overline{\gamma}}^{reg}]$, to eliminate the singularities from knots.

**Main**

**Theorem:**

## 9. Conclusions

**V**. (II) In order to relax this complicated structure, realizations of knots are produced on a four-dimensional Riemann manifold, which are formulated in the Riemann–Hilbert problem for the representation. (III) The monoidal transformations are applied at the knot intersection (singular) points, eliminating these from knots, thus producing the trivialization of the knots. The immediate consequence of these claims is the main theorem that the Zhang’s conjecture 1 proposed in [3] has been proved. The explicit expression for the resulting partition function Z have not been provided directly by the present procedure of the Riemann–Hilbert problem and the monoidal transformations. The partition function Z in a 4-fold integral form was presented in Equation (49) in [3] (see also Equation (24) in [4]) based on the Zhang’s two conjectures, which was proven to be correct in [8] by a Clifford algebra approach. The thermodynamical properties (including the free energy, the specific heat, the spontaneous magnetization, the spin correlation, the susceptibility, as well as the critical exponents) of the ferromagnetic 3D Ising model in the zero external field are derived explicitly in [3]. However, attention should be paid on using the present procedure to demonstrate rigorous formulation for the non-trivial knots’ components of the partition function Z. A subsequent paper, on the method of Riemann–Hilbert problem for Zhang’s conjecture 2, regarding to the generation of topological phases, will be published soon [81]. Furthermore, because the exact solution for the antiferromagnetic 3D Ising model with all the negative interactions but without frustration in the zero external field has the same formula as the exact solution obtained in [3,4,8], the results in [3,4,8] for the ferromagnetic 3D Ising model in the zero external field are suitable for the antiferromagnetic 3D Ising model without frustration.

_{3}_{B}T) > 0 (including the area of phase transitions, β

_{c}), where the non-trivial topological structures exist. It is clear now that the non-trivial topological structures (knots) contribute additional terms to the partition function and the physical properties (such as the free energy, the specific heat, the spontaneous magnetization, the spin correlation, the susceptibility, the critical exponents). Any approaches based on only local environments, such as conventional low-temperature expansions, conventional high-temperature expansions, Monte Carlo simulations, Renormalization Group, etc., are not exact, because these approximative approaches miss the contributions of knots. The conventional high-temperature expansions work only at/near infinite temperature β = 0, where only the trivial topological structure exists. Our procedures in [3,4,8] and the present work indicate that it is necessary to introduce an additional dimension (the time) to trivialize the knots and take into account their contributions, to achieve the exact solution of the 3D Ising models. The temperature–time duality in the 3D Ising model can be seen by inspecting the resemblance between the density operator in quantum statistical mechanics and the evolution operator in quantum field theory, with the mapping β = (k

_{B}T)

^{−1}→ it = τ [82,83,84]. In [84], Zhang and March pointed out that besides the Wick rotation, which represents the temperature as the imaginary time, we have to introduce also the time for the time average and for untying the knots (see also in [3,4,8] and this work). Therefore, one has to deal with the topological quantum field theory within a (3 + 2) or (4 + 1)-dimensional framework [85]. With Wick representation, the ε-expansions [86,87,88] start from four dimensions and do not account the non-trivial topological contributions, which are still in the approximative level. It can be improved by accounting the contributions of the non-trivial topological structure in the (3 + 2) or (4 + 1)-dimensional framework. Finally, we would like to notice that our work illustrates that the 3D Ising spin system can serve as a platform for describing a sensible interplay in between the physical properties of interacting many-body systems, algebra, topology, and geometry.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Theorem II

#### Appendix A.1. Construction of the Representation (Proof of (1) in Theorem II)

- (1)
- The case of basic type

_{1}to Q (respective P) is denoted by ${\gamma}_{1}^{(+)}$ (respective ${\gamma}_{1}^{(-)}$). We put ${\gamma}_{1}={\gamma}_{1}^{(+)}{\gamma}_{1}^{(-)}$.

- (2)
- The case of simple/multi-type

- (i)
- Choose a Riemann surface (Figure A4)

- (ii)
- Make a realization of γ by use of association of twisted type (Figure A5).

#### Appendix A.2. Construction of the Solutions (Proof of (2) in Theorem II)

- (1)
- The solution for the basic form of type I

- (2)
- The solution for the basic form of type II

- (3)
- The general type

_{1}, …a

_{n}} are the singularities arising from the knot of multi-type; (2) {b

_{1}, …, b

_{m}} are the singularities arising from the part of the given non-trivial knot. For the simplest sake, we write the sequence {a

_{n}, b

_{m}} as {c

_{k}}, k = 1, 2, …, n+m, and put ${\mathsf{\Gamma}}_{i}={\overline{\mathsf{\Gamma}}}_{i}=1$ for b

_{i}, we can rewrite them {c

_{k}}. $\{{\mathsf{\Gamma}}_{k},{\overline{\mathsf{\Gamma}}}_{k}\}$. Then, following the procedure above, we can formulate the Riemann–Hilbert problem and find the desired solution. The discussion is completely identical, and it may be omitted.

## Appendix B. Proof of Theorem III

**V**,

_{1}**V**, and

_{2}**V**. From the results in Section 4, there are knots of the following types: basic types I and II, product type (and adjoint type). In the following, we construct the trivializations for these basic types and combining the results, the desired trivialization is obtained for the product type.

_{3}_{i}, we have no information on knot structures. At each point, we associate the trivial matrix which is denoted by $\mathsf{\Gamma}({P}_{i})$ and $\overline{\mathsf{\Gamma}}({P}_{i})$, make extensions, for example, the extension is given at P

_{i}in the following manner and other cases are treated in a completely analogous manner (Figure A9).

^{2}(≅P

^{1}) which does not contribute to the partition functions. Thus, we may omit it.

_{2}, P

_{3}, the following diagram is obtained by monoidal transforms (Figure A11):

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**Figure 2.**A knot called basic form of type I, with the element $i{\mathsf{\Gamma}}_{i}{\overline{\mathsf{\Gamma}}}_{j}$ or $i{\overline{\mathsf{\Gamma}}}_{i}{\mathsf{\Gamma}}_{j}$.

**Figure 3.**A knot called basic form of type II, which can be represented as a braid with many crosses (k = 2n).

**Figure 4.**Schemes illustrate three of the braids in the transfer matrix

**V**connecting to the lattice points of the knot γ, while the circles of

_{3}**V**and

_{1}**V**are not shown for simplicity.

_{2}**Figure 6.**A 2-covering Riemann surface M

_{g}made by use of cut-segments where n is odd (n = 2m + 1).

**Figure 8.**The base Riemann surface M

_{g}in P × C as a covering space over P

^{1}in which the knot has a singularity of normal crossing.

**Figure 10.**The realization of knots on the Riemann surface. The dots represent the circle on the lower surface

**Figure 12.**The realization of the knot in Figure 11 on the Riemann surface.

b | 1 | $\overline{1}$ | C | $\overline{\mathit{C}}$ | |
---|---|---|---|---|---|

a | |||||

1 | 1 | 1 | C | C | |

$\overline{1}$ | $\overline{1}$ | $\overline{1}$ | $\overline{C}$ | $\overline{C}$ | |

C | C | C | 1 | 1 | |

$\overline{C}$ | $\overline{C}$ | $\overline{C}$ | $\overline{1}$ | $\overline{1}$ |

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**MDPI and ACS Style**

Suzuki, O.; Zhang, Z.
A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure. *Mathematics* **2021**, *9*, 776.
https://doi.org/10.3390/math9070776

**AMA Style**

Suzuki O, Zhang Z.
A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure. *Mathematics*. 2021; 9(7):776.
https://doi.org/10.3390/math9070776

**Chicago/Turabian Style**

Suzuki, Osamu, and Zhidong Zhang.
2021. "A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure" *Mathematics* 9, no. 7: 776.
https://doi.org/10.3390/math9070776