Why Zhang’s Proposed ”Exact Solution” of the

Three-Dimensional Ising Model is WrongJacques H. H. Perk

Oklahoma State University

Abstract:In 2007 Dr. Zhi-Dong Zhang conjectured a solution for the free energy and spontaneousmagnetization per site of the three-dimensional Ising model. With Dr. Norman H. March hehas advertised this solution as exact in several follow-up papers. We shall rigorously showthat the conjectured solution is not correct, however. For this purpose we shall provide adetailed proof of the corresponding specialization of a general theorem of the 1960s, onlyusing rather elementary mathematics. (See also arXiv:1209.0731 for further details.)

1

Why Zhang’s Proposed ”Exact Solution” of the

Three-Dimensional Ising Model is Wrong

Jacques H. H. PerkOklahoma State University

In 2007 Zhi-Dong Zhang published the very long paper:

[1] Z.-D. Zhang, Conjectures on the exact solution of three-dimensional (3D)simple orthorhombic Ising lattices, Philos. Mag. 87 (2007), 5309–5419[arXiv:cond-mat/0705.1045 (pp. 1–176)].

After being criticized in several comments, Zhang published many more papersmostly with Norman H. March. The last one contains all relevant references:

[2] Z.-D. Zhang and N. H. March, Mathematical structure of the three-dimen-sional (3D) Ising model, Chin. Phys. B 22 (2013) 030513 (15 pp.).

We shall see that there are many errors invalidating all claims.

2

Zhang’s results violate established series results,even in first nontrivial orders

To show this it is su�cient to restrict ourselves to the isotropic cubic Isingmodel (J1 = J2 = J3 = J , K = �J = J/kBT ). For that case Zhang expands his“putative” exact partition function per site in (A13) on page 5406 of [1] as

Z1/N = 2 cosh3 Kh1 +

722 +

878

4 +361348

6 + · · ·i, = tanhK. (1)

This di↵ers from the well-known high-temperature series given in (A12) of [1] as

Z1/N = 2 cosh3 Kh1 + 02 + 34 + 226 + · · ·

i. (2)

Zhang then claims that both results are correct, the first for finite temperatures,the second only for an infinitesimal neighborhood of � ⌘ 1/kBT = 0.

We shall rigorously show that the first result (1) is wrong and that series(2) has a finite radius of convergence. Our proof shall be rather elementary,compared to the proofs given in the 1960s.

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From Eq. (103) on page 5342 of [1] we obtain the “putative” low-temperatureexpansion of the spontaneous magnetization,

I = 1� 6x8 � 12x10 � 18x12 + · · · , x = e�2K . (3)

Some coe�cients of the well-known expansion are listed in Table 2 on page 5380of [1], implying

I = 1� 2x6 � 12x10 + 14x12 � · · · . (4)

The textbook derivation starts with a d-dimensional hypercubic lattice of Nsites, each site having 2d neighbors. There is one state with all spins up andenergy E+, N states with only one spin down and energy E+ + 4dJ , etc. Thus,

I =Z�

Z=

1 + (N � 2)x2d + · · ·1 + Nx2d + · · · = 1� 2x2d + · · · , 2d = 6. (5)

Only if the down spin is at the position of the spin operator we get a minus sign,or N � 2 = (N � 1)(+1) + (�1).

Again, Zhang’s result (3) is clearly wrong, displaying d = 4 behavior.

4

It has been shown in the 1960s that the high-temperature series of �f = limZ1/N

and all correlation functions on the cubic lattice have finite nonzero radius ofconvergence. By duality with the Ising model with 4-spin interactions on allfaces of the cubic lattice and at high temperatures, the low-temperature seriesfor the spontaneous magnetization I should also converge.

One cannot have two di↵erent expansions as given by Zhang.

To get out of this dilemma, Zhang (on pages 5381, 5383 and 5394 of [1]) comeswith the mathematically absurd claim that the old series are asymptotic withzero radius of convergence, whereas his “putative solution” is analytic with finiteradius of convergence. On pages 12 and 13 of [2] (and elsewhere) he claims thatthere is an (essential?) singularity at � = h = 0 and that this is due to the zerosof Z�1 and that Perk “went on perpetrating the fraud, discussing the singularityof” �f , not f .

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Irrelevant pole of fPerk proved (details later in this talk) that �f has an absolutely convergentseries, uniformly convergent in the thermodynamic limit, so that

�f =1X

i=0

ai�i, |�| < r.

(7)

Therefore,

f =a0

�+

1Xi=1

ai�i�1, 0 < |�| < r,

(8)

is a convergent Laurent series, totally equivalent to (7).

The pole at � = 0 has no significance, as �f is the relevant quantity fromstatistical mechanics point of view, entering the normalization Z = e�N�f forthe Boltzmann–Gibbs canonical distribution.

Also, note that in [1] Zhang expanded Z1/N = e��f , not f .

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Zeros of 1/Z are irrelevant

Zhang and March repeatedly claim the importance of the zeros of Z�1,with Z = zN the total partition function. However, unlike the complex Yang–Lee zeros of Z, the zeros of Z�1 are irrelevant:

For a finite number N of sites, Z is a finite Laurent polynomial in eK and onlycan become infinite when ReK = ±1, i.e. zero-temperature type limits.

For the infinite system, N !1, and finite K, the infinity of Z should be seen asjust a manifestation of the thermodynamic limit , in which z = Z1/N remainsfinite.

One can easily see that �f < 0 for K real, so that Z = e�N�f = 1 for all realK when N = 1. There is nothing special about the N = 1 zeros of Z�1.

Hence, Zhang and March cannot claim that the zeros of Z�1 play any role. Theyare there the same way for the one-dimensional Ising model, Z ⇡ (2 coshK)N ,and free Ising spins in field B, Z = (2 cosh�B)N , for N !1.

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Strategy: Analyticity of the correlation functionsand their thermodynamic limits

We first prove that the correlation functions of the Ising model on the periodiccubic lattice of size N = n3, have high-temperature series absolutely convergentwhen |�J | < r0, independent of n. The corresponding analyticity statement for�fN then follows using the elementary identity @(�fN )

@� = uN = 3Jh��0iN , with� and �0 nearest-neighbor spins.

As more and more coe�cients become independent of n, the analyticitythen carries over in the thermodynamic limit N !1.

Remark: From a well-known corollary to Bogoliubov’s inequality, we have

| log Tr eA � log Tr eB| kA�Bk, for A and B hermitian. (9)

Using this one can show that, for �J real and bounded, the free energy per sitefN converges uniformly to a limit f as the system size N becomes infinite in thesense of van Hove. (Ratio of number of boundary terms and volume tending tozero.) But this does not yet show analyticity.

8

Laurent Polynomial Lemma

The partition function ZN is a Laurent polynomial in e�J , i.e. polynomialin e�J and e��J . Therefore, �fN is singular only for the zeros of this Laurentpolynomial and for some cases with |e±�J | = 1. As ZN is a sum of positiveterms for real �J , ZN cannot have zeros on the real �J-axis .

It follows that all correlation functions hQ

l �liN are meromorphic functionswith poles only at the zeros of ZN .

Hence, DYl

�l

EN

=1X

i=1

ci(�J)i, |ci| < CN r�i, (10)

with r the absolute value of the zero closest to �J = 0 and CN some positiveconstant.

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Lemma (M. Suzuki, 1965) for B = 0

⌧ mYi=1

�ji

�N

=1m

mXk=1

⌧✓ mYi=1i6=k

�ji

◆tanh

✓�J

Xl nn jk

�l

◆�N

,

(11)

where j1, . . . , jm are the labels of m spins and l runs through the labels of thesix spins that are nearest neighbors of �jk . (The lemma also holds withoutaveraging over k.)

The proof simply follows summing over spin �jk in the numerator of theexpectation value, i.e.,

X�jk

=±1

�jke�J

Pl nn jk

�jk�l = tanh

✓�J

Xl nn jk

�l

◆ X�jk

=±1

e�J

Pl nn jk

�jk�l

. (12)

M. Suzuki, Phys. Lett. 19 (1965), 267–268, Intern. J. Mod. Phys. B 16 (2002), 1749–1765.

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Expanding tanh Lemma

tanh✓

�J6X

l=1

�l

◆= a1

X(6)

�l + a3

X(20)

�l1�l2�l3 + a5

X(6)

�l1�l2�l3�l4�l5 ,

(13)

where the sums are over the 6, 20, or 6 choices of choosing 1, 3, or 5 spins fromthe given �1, . . . ,�6. It is easy to check that the coe�cients ai are

a1 =t(1 + 16t2 + 46t4 + 16t6 + t8)

(1 + t2)(1 + 6t2 + t4)(1 + 14t2 + t4), a3 =

�2t3

(1 + t2)(1 + 14t2 + t4),

a5 =16t5

(1 + t2)(1 + 6t2 + t4)(1 + 14t2 + t4), t ⌘ tanh(�J). (14)

The poles of the ai are at t = ±i, t = ±(p

2 ± 1)i, and t = ±(p

3 ± 2)i. Itcan also be verified, e.g. expanding the ai in partial fractions, that the seriesexpansions of the ai in terms of the odd powers of t alternate in sign and convergeabsolutely as long as |�J | < arctan(2�

p3) = ⇡/12.

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Proof of Expanding tanh Lemma

Clearly, tanh��J

P6l=1 �l

�can be expanded as done. Replacing all six

spins, �l by ��l, shows that no terms with an even number of spins occur. Also,permutation symmetry allows only three di↵erent coe�cients.

Multiplying with one, three, or five spins �l and then summing over all26 = 64 spin states, one finds the coe�cients a1, a3, and a5, which can the beexpanded in partial fraction expansions,

a1,5 =124

⇣ p1t

1 + (p1t)2+

p2t

1 + (p2t)2⌘±p

28

⇣ p3t

1 + (p3t)2+

p4t

1 + (p4t)2⌘

+13

p5t

1 + (p5t)2,

a3 =124

⇣ p1t

1 + (p1t)2+

p2t

1 + (p2t)2⌘� 1

6p5t

1 + (p5t)2,

p1,2 = 2 ±p

3, p3,4 =p

2 ± 1, p5 = 1, (p1p2 = p3p4 = 1).

(15)

(16)

The remaining statements of the lemma follow from these expansions.

12

Uniform convergence for finite NExpanding the tanh Suzuki’s lemma replaces any even correlation by a linearcombination of even correlations, with coe�cients given by the ai. As, for fixed|t| ⌘ x, each |ai| is maximal for imaginary t = ix, we find

s ⌘ 6|a1| + 20|a3| + 6|a5| 2x(3� x2)(1� 3x2)

(1� x2)(1� 14x2 + x4). (17)

It easily shown that the sum of the absolute values of all coe�cients s < 1 for

|t| < (p

3�p

2)(p

2� 1) = 0.131652497 · · · , or

|�J | < arctan[(p

3�p

2)(p

2� 1)] = 0.130899693 · · · . (18)

This determines a lower bound on the radius of convergence:Repeat applying Suzuki’s lemma on all correlations that are not h1i = 1.

After I iterations, a given even correlation is than split into an explicitly givensum bounded by s + s2 + · · · + sI and a remainder sum with correlations leftfor further iteration, higher order in t or �J . As we saw that each correlationcan have at most poles given by the complex zeros of Z, we conclude that theremainder is bounded by sI+1 + sI+2 + · · ·.

13

Uniform convergence for infinite N

Theorem: hQm

i=1 �jiiN converges to a unique limit as N !1 for |t| < 2�p

3,defined by its series, as more and more coe�cients become independent of Nwith increasing N and the remainder tends to zero uniformly.

The above iteration process generates new correlations with the range ofthe positions ji of the spins extended by one in a given direction. As long as wedo not go around a cycle (periodic boundary condition) of the 3-torus, we donot notice any N -dependence.

More precisely: Let d be the largest edge of the minimal parallelepipedcontaining all sites j1, . . . , jm. Then the coe�cient of tk with k < n � d forthe lattice with N = n3 sites equals the corresponding coe�cient for larger N ,including the one for N = 1. It takes at least n � d iteration steps to noticethe finite size of the lattice.

14

Theorem for reduced free energyand its thermodynamic limit

The reduced free energy �fN for arbitrary N and its thermodynamic limit �f areanalytic in �J for su�ciently high temperatures. They have series expansionsin t or �J with radius of convergence bounded below by ( suzidn) and uniformlyconvergent for all N including N = 1. The first n�1 coe�cients of these seriesfor N = n3 equal their limiting values for N = 1.

Proof: To prove analyticity of �f in terms of � at � = 0 it su�ces to study theinternal energy per site or the nearest-neighbor pair correlation function, as

uN =1NhHN iN =

@(�fN )@�

= �3Jh�0,0,0�1,0,0iN . (19)

Here �0,0,0 and �1,0,0 can be any other pair of neighboring spins. The proof thenfollows from the theorem for correlation functions and integrating the series foruN , using ZN |�=0 = 2N , implying lim�!0 �fN = � log 2.

15

The first nontrivial coe�cient

Apply Suzuki’s lemma once to h�0,0,0�1,0,0i, we find

h�0,0,0�1,0,0i = a1 h1i+ O(t2) = t + O(t2) = �J + O(�2). (20)

Hence,

�f = � log 2�Z �

03Jh�0,0,0�1,0,0id� = � log 2� 3

2(�J)2 + O(�3).

(21)

This and the next few terms so obtained agree with the usual series expansion(2), but disagree with Zhang’s “putative” exact result.

The only possible conclusion is that the conjectured answers of Zhang [1,2]are wrong and we shall next see more reasons why.

16

No obvious choice of weight functions

One problem with conjecture 2 [1,2] is that there is no obvious choice for theweight functions. In (49) on page 5325 of [1] Zhang writes

N�1 lnZ = ln 2 +1

2(2⇡)4

Z ⇡

�⇡

Z ⇡

�⇡

Z ⇡

�⇡

Z ⇡

�⇡ln

⇥cosh 2K cosh 2(K0 + K00 + K000)

� sinh 2K cos!0 � sinh 2(K0 + K00 + K000)(wx cos!x

+ wy cos!y + wz cos!z)⇤d!0d!xd!yd!z, (22)

with K = K0 = K00 = K000 for the isotropic cubic lattice.

Even though this is a result of flawed assumptions, it is an integral transform thatcould give the right answer with a suitable choice of weight functions wx, wy, wz.Zhang’s wrong “putative” result comes from the choice wx = 1, wy = wz = 0.In appendix A of [1] Zhang reversely engineers an other truncated series choicederived from known coe�cients of the Domb–Sykes high-temperature series.There is no more information than is in the limited series results provided byothers, so that this is not an exact result.

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Conjecture 1 is manifestly wrong

The original paper [1] has an error in the application of the Jordan–Wignertransformation pointed out by Perk. This error has only been corrected explicitlyin [2], which makes it easy to pinpoint the error with Conjecture 1:

It is well-known that in the spinor representation of the orthogonal groupseach element g can be written as a “fermionic Gaussian” of the form

g = exp✓

12

Xi

Xj

Aij�i�j

◆, Aji = �Aij , (23)

with Cli↵ord algebra elements satisfying �i�j + �j�i = 2�ij and antisymmetriccomplex coe�cients Aij . The spinor representation has been used in the Isingcontext first by Kaufman in 1949.

The closure property of Lie groups says that any product or inverse ofelements of this form is again of the same fermionic Gaussian form.

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Remark: Lie group and Lie algebra

The group elements g act on the � ’s as

�k �! g�kg�1. (24)

Choosing infinitesimal

g = 1 +"

2

Xi

Xj

Aij�i�j + O("2), (25)

we find from (24) in O("2), that the corresponding Lie algebra action is12

Xi

Xj

Aij�i�j , �k

�=

Xi

Aik�i , (26)

showing that the infinitesimal action is through multiplication by antisymmetricmatrices, the generators of rotations.

Therefore, the g’s indeed form a representation of a rotation group.

19

Transfer matrix

It is well-known that the free energy per site in the thermodynamic limit doesnot depend on boundary conditions. Therefore, the Hamiltonian (1) in [1] canbe rewritten using the scew boundary conditions of Kramers and Wannier as

��H =nX

⌧=1

mlXj=1

hKs(⌧)

j s(⌧+1)j + K0s(⌧)

j s(⌧)j+1 + K00s(⌧)

j s(⌧)j+m

i. (27)

For this purpose we have made the change of notation

s(⌧)⇢,� ⌘ s(⌧)

j , j ⌘ ⇢ + (m� 1)�,mX

⇢=1

lX�=1

=mlXj=1

, (28)

where ⌧ = 1, · · · , n; ⇢ = 1, · · · ,m; � = 1, · · · , l.This leads to the transfer matrix T = V3V2V1 in [2], with

H.A. Kramers and G.H. Wannier, Phys. Rev. 60 (1941) 252–262.

20

V3 = exp✓� iK00

mlXj=1

�2j

j+m�1Yk=j+1

i�2k�1�2k

��2j+2m�1

◆, (29)

V2 = exp✓� iK0

mlXj=1

�2j�2j+1

◆, V1 = exp

✓iK⇤

mlXj=1

�2j�1�2j

◆, (30)

compare (15), (16) and (17) of [2], with n replaced by m. Clearly, (29) is not ofthe fermionic Gaussian form and, therefore, not an element of the group.

At this point, Zhang introduces a fourth dimension, stacking o copies of themodel. Without changing the free energy per site, one can connect the copiesto give (29) and (30) with the upper bounds of the sums ml replaced by mlo.

Next, Zhang made the absurd conjecture that multipying V30V2

0V10 so obtained

by

V40 = exp

⇢� iK000

mloXj=1

�2j�2j+1

�, K000 =

K0K00

K,

as given in (18) and (19) in [2], miraculously produces a rotation group element.The argumentation in [1] for the form of K000 also makes no sense.

21

Appendix: Other issues

Zhang and March also falsely claim that setting � = 1 in 1960’s references is aloss of generality, losing T = 1. On the contrary, as �f is only a function ofK = �J , this is no problem. Having J ⌘ K and choosing a fixed new J and anew � = J/J 6⌘ 1, we can write J = K = �J . Thus we recover the general casewith both a fully variable � (including � = 0) and a new J (omitting the bars).

Next, Zhang and March fail to realize that K��0(X,T ) [in Phys. Lett. A 25(1967), 493–494] vanishes for � = 0. The cited inequality does not fail for � = 0.

Finally, only in two dimensions is the conformal group infinite-dimensional, sothat there is further inconsistency using Virasoro algebra in [2] and cited papers,beyond the fact that these papers build on an erroneous solution of the 3D Isingmodel. That Zhang and March write Re|ei�i |, the real part of a positive realnumber, is objectionable too.

22

Acknowledgments

The author thanks Professor M. E. Fisher for a number of useful suggestions.The research was supported in part by NSF grant PHY 07-58139.

The End

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