Comment on Zhang, D. Exact Solution for Three-Dimensional Ising Model. Symmetry 2021, 13, 1837

Abstract

We show that Zhang Degang’s claimed solution of the three-dimensional Ising model has fatal irreparable errors.

In [1], Zhang Degang claims to have solved the free energy per site of the three-dimensional Ising model with screw boundary conditions. This claim evolved in earlier versions [2], in part due to my referee reports on them, in which I stated that I had found that the claimed result failed the well-known high-temperature series test for the thermodynamic free energy. This failure was confirmed in version 4 of [2], leading to the additional erroneous claim of the dependence on boundary conditions of the free energy per spin in the thermodynamic limit.

However, this free energy per site in the thermodynamic limit is to be independent of boundary conditions [3,4]. This follows from the Peierls–Bogolyubov inequality, which implies that the difference of total free energies is bounded by the norm of the difference of their Hamiltonians,

$$|F\left[{\mathcal{H}}_2\right]-F\left[{\mathcal{H}}_1\right]|\le \left|\right|{\mathcal{H}}_2-{\mathcal{H}}_1\left|\right|,F\left[{\mathcal{H}}_i\right]\equiv -{\beta }^{-1}log\mathrm{Tr}{\mathrm{e}}^{-\beta {\mathcal{H}}_i},(i=1,2).$$

(1)

As $ln$ (or $2ln$ according to Figure 1 in [1]) bonds $J_1$ are moved from periodic to screw boundary conditions, the right-hand side of (1) is bounded by $2ln|J_1|$ (or $4ln|J_1|$). Per site, we must divide by $lmn$ (or $2lmn$), so the free energies per site differ by at most $2|J_1|/m$, which becomes zero in the thermodynamic limit $l,m,n\to \infty$. Hence, the results per atom for periodic and screw boundary conditions, and also (44) and (45) in [1], should be equal.

On the middle of page 5 of [1], we read ${\mathcal{A}}_{p,s}\equiv A_{p,s}$, equivalent because they commute and have the same eigenvectors and eigenvalues. However, equating this “≡” and “=” is a serious error, as then ${\sigma }^z\equiv -{\sigma }^z$ would imply ${\sigma }^z=-{\sigma }^z$, for example. More generally, we should expect the common eigenvalues of ${\mathcal{A}}_{p,s}$ and $A_{p,s}$ to be distributed differently over the common eigenvectors, so that ${\mathcal{A}}_{p,s}\ne A_{p,s}$ instead of being equal.

This error is present in (16) in [1], where we may replace ${\sum }_{p=1}^m{\mathcal{A}}_{p,1}={\sum }_{p=1}^m{A}_{p,1}$ by ${\mathcal{A}}_{p,1}=A_{p,1}$. (As m is arbitrary, the equality of the two sums in (16) is equivalent to the equality of their summands.) However, by (15) and the text below it in [1],

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{p,1}& =& {\mathcal{L}}_{1,2}^p={\sigma }_p^z{\sigma }_{p+m}^z\equiv A_{p,1}=L_{1,2}^p\hfill \\ & =& L_{p,p+m}={\sigma }_p^z{\sigma }_{p+1}^x\cdots {\sigma }_{p+m-1}^x{\sigma }_{p+m}^z,\hfill \end{array}$$

(2)

after both reconstructing missing definitions from the text below (13) and using (3) in [1]. Indeed, ${\sigma }_p^z{\sigma }_{p+m}^z$ and ${\sigma }_p^z{\sigma }_{p+1}^x\cdots {\sigma }_{p+m-1}^x{\sigma }_{p+m}^z$ commute and have the same eigenvectors and eigenvalues, but these $\pm 1$ eigenvalues are distributed differently over the common eigenvectors. Therefore, equating ${\mathcal{A}}_{p,1}=A_{p,1}$, as is used in (17) of [1], is wrong.

Furthermore, comparing, in [1], the first line of (17) with (2) for $V=V_1{V}_2{V}_3$, we see that Zhang has set the two V’s equal for all temperatures, implying the erroneous equality of $H_y={\sum }_{\tau =1}^{mn}{\sigma }_{\tau }^z{\sigma }_{\tau +m}^z$ and Onsager’s $A_m={\sum }_{\tau =1}^{mn}{\sigma }_{\tau }^z\left({\prod }_{j=1}^{m-1}{\sigma }_{\tau +j}^x\right){\sigma }_{\tau +m}^z$, identifying $s={\sigma }^z$ and $C={\sigma }^x$ in (45) and (56) of [5]. In fact, it is the typical error in most incorrect solutions of 3D Ising. Since 1975, as a referee, I have rejected several manuscripts, in which the 3D Ising model was incorrectly reduced in a somewhat similar way to free fermions and I have commented on one other such work [6]; see also Section 6.2 of [7].

There are more reasons to see that [1] is flawed. The formula for the critical temperature (32),

$$sinh\left(2\beta J\right)sinh(2\beta J_1+2\beta J_2)=1,\beta =1/k_{\mathrm{B}}T,$$

(3)

can lead to three different critical temperatures by just rotating the lattice. Indeed, using ($J_1$, $J_2$, J) as a permutation of (0.5, 1.0, 1.5), one obtains three critical temperatures, with ${\beta }_c=1/k_{\mathrm{B}}{T}_c$ being 0.3503982204, 0.3046889317, or 0.2937911957, depending on J being 0.5, 1.0, or 1.5. According to [8], however, these values are three upper bounds on the true ${\beta }_c$, three lower bounds on the true $T_c$, and (3) only provides the critical point asymptotically in the limit $J\gg J_1,J_2\ge 0$.

When $J_1=J_2=J=1$ one finds 0.3046889317, thus disagreeing with about every result in the literature. The best value to date may be 0.221654626(5) [9]. Almost all calculations, using series, Monte Carlo, finite-size extrapolations, renormalization group, etc., point in the direction of 0.22. Only very few calculations provide different numbers and one can quickly locate obvious errors in those calculations.

Furthermore, the critical exponents do not agree at all with the best values in the literature, see, e.g., [10,11,12] and the references cited. The critical exponent $\alpha =0$ found disagrees with $\alpha =0.11\cdots$ supported by the most reliable estimates in the literature. The relation with the 2D Ising claimed would seem to imply $\beta =1/8$ and $\eta =1/4$, which are far outside the values obtained using a series and Monte Carlo.

In conclusion, the errors in [1] cause the results to be wrong.