Abstract
The mean-field approximation predicts a phase transition in every dimension. This paper asks what settles how far it errs—the answer is the dimension alone. No dimension enters the approximation, and the size of the error is a function of the dimension nonetheless. No new mathematical theorem and no new law is claimed.
First, in one dimension it errs without limit. Exactly, $T_c=0$ and there is no transition; mean field says $kT_c/J=\mathbf{2}$—no ratio can be formed(Section 2).
Second, this is the core of the paper. The exact two-dimensional value (Onsager) is $kT_c/J=\mathbf{2.269185}$ and mean field gives $\mathbf{4}$—$\mathbf{76.27\%}$ too high(Section 3).
Third, raising the dimension shrinks the error. $\mathbf{32.99\%}$ in three dimensions and $\mathbf{19.76\%}$ in four—falling monotonically(Section 3).
Fourth, and yet it never reaches zero. The ratio runs $1.76275\to1.32993\to1.19755$ towards $1$, and never attains it at any finite $d$(Section 4).
Fifth, the exponents, on the other hand, become exact at $d\ge4$. Above the upper critical dimension the critical exponents agree with the mean-field values—one and the same approximation becomes exact for one quantity and errs to the last for another(Section 5).
Sixth, for continuous symmetry it is worse. By Mermin-Wagner there is no spontaneous breaking for $d\le2$, and mean field says it happens in every dimension(Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed—the mean-field approximation, Onsager’s exact solution, that the upper critical dimension is $4$, and the Mermin-Wagner theorem are all standard. No statistical mechanics is built—what is used is one ratio and the comparison of four values. No exact solution is derived—Onsager’s $2/\ln(1+\sqrt2)$ is cited, and the three- and four-dimensional values are estimates from numerical work. The renormalisation group is not entered—Papers 117 and 120 treat it. This paper only sets out the size of the error. No critical exponent is computed—Paper 117 treats mean field as the naive dimensional counting that gives $\eta=0$ and writes the upper critical dimension $4$ from the power counting of the coupling. This paper does not enter that discussion of exponents and looks only at the value of $T_c$. Orders of limits are not treated—Paper 245 treats the order of limits in the Curie-Weiss model and states explicitly that it treats neither critical exponents nor the renormalisation group. This paper concerns the size of the error, not the order. Mermin-Wagner is not proved—only its conclusion is cited. The $8$ occurrences of Mermin in earlier papers (11, 41, 185) are quantum measurement and the classification of defects, a different work by the same man. The upper critical dimension of percolation is not conflated with this—the $6$ of Paper 264 belongs to percolation and the $4$ here to the Ising model. Relation to earlier papers: Paper 117 separated the four ways in which scale invariance fixes an exponent—this paper looks at the transition temperature rather than the exponents. Paper 120 showed that dimensional analysis fails in two ways—this paper counts the size of the failure, dimension by dimension. Paper 245 showed that two limits have no answer until the order is written—this paper looks at the same mean-field model from the side of accuracy rather than order. Paper 264 wrote that the threshold belongs to the lattice and the exponents to the dimension—this paper writes that $T_c$ and the exponents follow the dimension differently. The same shape of division. Paper 190 measured rare on a logarithmic scale—this paper writes in percentages and ratios. What is added is counting the error as $76.27\%$, $32.99\%$ and $19.76\%$ by dimension, writing that no ratio can be formed in one dimension, setting out $1.76275\to1.19755$ to show approach without attainment, and writing the asymmetry that the exponents become exact at $d\ge4$ while $T_c$ never does.
1. Notation and Placement
Setting: take the Ising model. Place a spin of $\pm1$ at each lattice site and sum $-J$ times the product over neighbouring pairs. On a simple cubic lattice in $d$ dimensions the number of neighbours, the coordination number, is $z=2d$.
The mean-field approximation: replace the field one spin feels by the average of its surroundings. The transition temperature then comes out as $kT_c/J=z=2d$—it sees only the number of neighbours.
Question: how far does this approximation err?
The answer is settled by the dimension alone. No “dimension” enters the approximation—only the coordination number does. And still the error is a function of the dimension.
A caution: Mermin in this paper means the Mermin-Wagner theorem. The $8$ occurrences in earlier papers are quantum measurement and the classification of defects, a different work by the same man—the two are not conflated.
2. In One Dimension It Errs without Limit
Check 2.1: solving the one-dimensional Ising chain exactly, no magnetisation stands at any finite temperature.
| $kT_c/J$ | |
|---|---|
| Exact (one dimension) | $\mathbf{0}$ (no transition) |
| Mean field ($z=2$) | $\mathbf{2}$ |
| Ratio | cannot be formed |
Observation 2.2 this is not “poor accuracy”. It says that something is there which is not—a kind of error that cannot be written as a percentage.
Observation 2.3 the reason can be given in one line. In one dimension the cost of flipping a single place is finite, so order is destroyed at any temperature however low—and mean field does not see that “single place”.
3. The Core: Counting the Error by Dimension
Check 3.1: the exact two-dimensional value was given by Onsager ($1944$), $kT_c/J=2/\ln(1+\sqrt2)$.
| $d$ | $z=2d$ | Mean field | Exact (or numerical) | Excess |
|---|---|---|---|---|
| $1$ | $2$ | $2.0000$ | $0.000000$ | cannot be formed |
| $2$ | $4$ | $4.0000$ | $\mathbf{2.269185}$ | $\mathbf{76.27\%}$ |
| $3$ | $6$ | $6.0000$ | $4.511500$ | $\mathbf{32.99\%}$ |
| $4$ | $8$ | $8.0000$ | $6.680300$ | $\mathbf{19.76\%}$ |
Observation 3.2 the error shrinks monotonically. $76.27$ to $32.99$ to $19.76$ per cent—the higher the dimension, the better mean field does.
Observation 3.3 the separator—what has to be changed for the error to change. The dimension alone. Neither the strength of the coupling, nor the shape of the lattice, nor the size of the spin has been moved—and no dimension appears in the mean-field expression, while the error is a function of it.
Observation 3.4 the reason is that mean field sees no fluctuation at all. The higher the dimension the more neighbours there are, and fluctuations average away—the approximation improves not because it becomes right, but because what it ignored becomes small.
4. And Yet It Never Reaches Zero
Check 4.1: set out the ratio, mean field over exact.
| $d$ | Mean field over exact |
|---|---|
| $2$ | $\mathbf{1.76275}$ |
| $3$ | $\mathbf{1.32993}$ |
| $4$ | $\mathbf{1.19755}$ |
| $d\to\infty$ | $1$ (in the limit only) |
Observation 4.2 it approaches $1$ and never arrives at any finite $d$. Mean field is exact for $T_c$ only in infinite dimension—in any real dimension it is out, however far one goes.
Observation 4.3 this says the same thing as the $76.27\%$ of Section 3 in another notation. Both express “approaching without arriving”.
5. The Exponents Become Exact at $d\ge4$ and $T_c$ Never Does
Observation 5.1 the upper critical dimension is $4$. For $d\ge4$ the critical exponents agree with the mean-field values (up to logarithmic corrections).
Check 5.2: set the two quantities out by how they follow the dimension.
| Quantity | At $d\ge4$ | At $d\to\infty$ |
|---|---|---|
| Critical exponents | become exact | exact |
| Transition temperature $T_c$ | still out by $19.76\%$ | exact only there |
Observation 5.3 one and the same approximation becomes exact for one quantity at a finite dimension and errs to the last for the other. The phrase “the dimension at which mean field becomes right” names two different things—$4$ if one means the exponents, and infinity if one means $T_c$.
Observation 5.4 Paper 117 treated the side of the exponents—placing mean field as the naive dimensional counting that gives $\eta=0$, and writing the upper critical dimension $4$ from the power counting of the coupling. This paper is on the side of $T_c$, and that the two sides give different answers is the content of this section.
Observation 5.5 Paper 264 wrote, for percolation, that the threshold belongs to the lattice and the exponents to the dimension. This paper writes, for the Ising model, that $T_c$ and the exponents follow the dimension differently—the same shape of division appearing in another phenomenon.
6. For Continuous Symmetry It Is Worse
Observation 6.1 Ising spins take the two values $\pm1$, that is, they are discrete. Make the spin continuously rotatable (Heisenberg or XY) and matters change.
Check 6.2: set the Mermin-Wagner conclusion beside the mean-field prediction.
| Dimension | Spontaneous breaking of continuous symmetry | Mean field says |
|---|---|---|
| $d=1$ | does not occur | it occurs |
| $d=2$ | does not occur | it occurs |
| $d=3$ | occurs | it occurs |
Observation 6.3 with discrete symmetry (Ising) order stands in two dimensions, and with continuous symmetry it does not. Mean field sees this distinction not at all—it returns the same answer either way.
Observation 6.4 so how far mean field errs depends not only on the dimension but on the type of the symmetry. The table of Section 3 is for discrete symmetry, and for continuous symmetry it is worse—a point to state honestly.
7. Roll Call and Closing
- Unbounded error in one dimension(Check 2.1): exactly $0$, mean field $2$.
- An error no percentage can express(Observation 2.2): it says something is there which is not.
- $76.27\%$ too high in two(Check 3.1): $4$ against $2.269185$.
- $32.99\%$ in three, $19.76\%$ in four(Check 3.1): shrinking monotonically.
- The separator is the dimension alone(Observation 3.3): which does not appear in the expression.
- Approaching without arriving(Check 4.1): $1.76275\to1.19755$.
- The exponents are exact at $d\ge4$(Check 5.2): $T_c$ only at infinity.
- The two sides differ(Observation 5.3): $4$, and infinity.
- Worse for continuous symmetry(Check 6.2): it says order stands where none does.
Closing: the mean-field expression contains no dimension at all. What it contains is the coordination number $z$—and still the error was a function of the dimension alone. In one dimension it says something is there which is not; in two it is $\mathbf{76.27\%}$ too high, in three $32.99\%$, in four $19.76\%$—shrinking monotonically and never reaching zero at any finite dimension. And the critical exponents do become exact at $d\ge4$—one and the same approximation becomes exact for one quantity at a finite dimension and only at infinity for the other. For continuous symmetry it is worse still, saying that order stands at $d\le2$ where none does. One thing separates them—writing down what one means by “mean field becomes right”. Write it down, and the $4$ is seen to belong to the exponents and not to $T_c$. Do not write it down, and one reads mean field as having become entirely right in four dimensions.
References
- Onsager, L. (1944). Crystal Statistics I: A Two-Dimensional Model with an Order-Disorder Transition. Physical Review, 65(3).
- Mermin, N. D. and Wagner, H. (1966). Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models. Physical Review Letters, 17(22).
- Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift fuer Physik, 31(1).
- Goldenfeld, N. (1992). Lectures on Phase Transitions and the Renormalization Group. Reading: Addison-Wesley.
- Baxter, R. J. (1982). Exactly Solved Models in Statistical Mechanics. London: Academic Press.
- Ferrenberg, A. M., Xu, J. and Landau, D. P. (2018). Pushing the Limits of Monte Carlo Simulations for the Three-Dimensional Ising Model. Physical Review E, 97(4).
- Yamagishi, Y. (2026). Scale Invariance Fixes an Exponent — in Four Different Ways [Paper 117]. Zenodo. 10.5281/zenodo.22005515
- Yamagishi, Y. (2026). Dimensional Analysis Fails in Two Ways [Paper 120]. Zenodo. 10.5281/zenodo.22036433
- Yamagishi, Y. (2026). “Rare” Is Measured on a Logarithmic Scale [Paper 190]. Zenodo. 10.5281/zenodo.22106182
- Yamagishi, Y. (2026). Two Limits Have No Answer Until the Order Is Written [Paper 245]. Zenodo. 10.5281/zenodo.22159739
- Yamagishi, Y. (2026). Inside One Phenomenon Live a Number That Moves with the Lattice and a Number That Does Not [Paper 264]. Zenodo. 10.5281/zenodo.22169720
Reference
Yamagishi, Y. (2026). Mean Field Does Not See the Dimension, and Yet the Dimension Alone Settles How Far It Errs [Paper 271]. YOI: https://www.yamagishiyuuki.com/yoi/271
DOI: 10.5281/zenodo.22169925
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 271. This paper is a preprint and has not been peer reviewed.