Mean-Field (Weiss) Theory of the Ising Model

The Ising model is trivial to write down but, beyond one dimension, notoriously hard to solve exactly. Mean-field theory — in its oldest guise, Pierre Weiss's molecular-field theory of ferromagnetism — is the first fully analytical treatment of its phase transition, and it remains the reflex first calculation for any new ordering problem. Its single idea is to stop tracking correlations between spins: each spin is treated as independent, feeling not its neighbours individually but only their thermal average, an internal "molecular field" that the spin itself helps to create. An interacting many-body problem collapses into a one-spin problem closed by a single transcendental equation. Concretely, replace the neighbour sum $\sum_j S_j$ acting on a spin by $z\langle S\rangle=zm$ , where $z$ is the coordination number and $m$ the magnetization per spin. Each spin then sits in an effective field $h_{\mathrm{eff}}=Jzm+h$ , and its thermal average must reproduce the very $m$ we assumed: the self-consistency condition $m=\tanh[\beta(Jzm+h)]$ . Graphically this is the intersection of the line $y=m$ with the S-curve $y=\tanh(\beta Jz\,m)$ : at high temperature the only crossing is $m=0$ (disorder), but once the curve's initial slope $\beta Jz$ exceeds one, two symmetric nonzero solutions appear. Spontaneous magnetization is born exactly when that slope equals one, fixing the critical temperature $k_BT_c=Jz$ . Near $T_c$ the theory predicts the classical (Landau) critical exponents $\beta=\tfrac12$ , $\gamma=1$ , $\delta=3$ , $\alpha=0$ , independent of dimension — the hallmark of the mean-field universality class. Those exponents are exact above the upper critical dimension $d_c=4$ and for infinite-range (Curie–Weiss) couplings, but wrong below. Most tellingly, mean-field theory predicts a finite-temperature transition even in the one-dimensional chain, where the exact answer is $T_c=0$ : with $z=2$ it announces $k_BT_c=2J$ , a transition that does not exist. This spurious 1D order is the theory's most instructive failure, and it is precisely what fluctuations, the Ginzburg criterion, and the renormalization group were built to repair.

Key takeaways

Mean-field (Weiss) theory replaces the fluctuating field from a spin's neighbours by its thermal average, so each spin feels an effective field $h_{\mathrm{eff}}=Jzm+h$ with $m$ the magnetization and $z$ the coordination number.
The magnetization obeys the self-consistent Weiss equation $m=\tanh[\beta(Jzm+h)]$ , whose nonzero solution appears below $k_BT_c=Jz$ .
Near $T_c$ the classical exponents follow: $\beta=\tfrac12$ , $\gamma=1$ , $\delta=3$ , $\alpha=0$ , obeying Rushbrooke ( $\alpha+2\beta+\gamma=2$ ) and Widom ( $\gamma=\beta(\delta-1)$ ) as equalities, independent of dimension.
The same result follows variationally (Gibbs–Bogoliubov bound), so the mean-field free energy is a rigorous upper bound on the true free energy.
Mean-field theory is exact for infinite-range couplings and above $d_c=4$ , but fails in low dimensions — it falsely predicts a transition at $k_BT_c=2J$ in $d=1$ , where the exact $T_c=0$ .
Key idea: Replace a spin's interaction with its neighbours by its interaction with the average magnetization, $J\sum_j S_iS_j\to(Jzm)S_i$ , turning the coupled problem into independent spins in a self-consistent effective field.

The core insight

Replace a spin's interaction with its neighbours by its interaction with the average magnetization, $J\sum_j S_iS_j\to(Jzm)S_i$ , so each spin becomes an independent two-level system in the self-consistent Weiss field $h_{\mathrm{eff}}=Jzm+h$ ; the magnetization then satisfies $m=\tanh[\beta(Jzm+h)]$ , ordering below $k_BT_c=Jz$ with the classical exponents $\beta=\tfrac12,\gamma=1,\delta=3,\alpha=0$ .

Where this lives today

Mean-field theory is the universal first pass and the fixed reference against which every fluctuation correction is measured. The high-temperature susceptibility $\chi=C/(T-T_c)$ is the experimentally observed Curie–Weiss law for paramagnets above their Curie point; plotting $\chi^{-1}$ versus $T$ gives a straight line whose intercept estimates $T_c$ , with deviations near $T_c$ signalling the onset of true critical fluctuations. Through the lattice-gas/Ising mapping, the same self-consistent equation becomes the Bragg–Williams theory of order–disorder transitions in alloys such as $\beta$ -brass, and the van der Waals equation of state for the liquid–gas critical point — the reason diverse fluids show near-classical behaviour far from $T_c$ . Above the upper critical dimension $d_c=4$ mean-field exponents are the exact answer, and they are the starting point of the $\epsilon=4-d$ expansion of the renormalization group, whose whole purpose is to compute the corrections mean-field theory misses.

The physical picture

Imagine one spin buried in a crystal of many others. It does not resolve each neighbour separately; it feels the net pull of the surrounding magnetized medium. Weiss's leap in 1907 — decades before the Ising model was written down — was to declare that net pull simply proportional to the average magnetization, a "molecular field." But that field is itself produced by the spins' response, so the value of $m$ cannot be chosen freely: the field you assume must generate exactly the magnetization that creates it. Solving that loop is the whole theory.

Figure

Graphical self-consistency: m = tanh(βJzm) — one root above T_c, three below

Read off the picture: the self-consistency equation $m=\tanh(\beta Jz\,m)$ is the crossing of the straight line $y=m$ with the S-shaped $y=\tanh(\beta Jz\,m)$ . At high $T$ the tanh is shallow and meets the line only at the origin — the disordered state. As $T$ drops, the slope at the origin, $\beta Jz$ , climbs past one; the curve bulges above the line and two symmetric nonzero crossings appear. The birth of spontaneous magnetization is exactly the moment the initial slope hits one, giving $k_BT_c=Jz$ . Why does this fail in low dimensions? Mean-field theory replaces a fluctuating quantity by its mean — a superb move when each spin couples to many neighbours (large $z$ , high $d$ ), because the law of large numbers then tames fluctuations. In a one-dimensional chain a spin has only two neighbours; a single thermal flip can unzip order along the whole line, an effect mean-field theory is structurally blind to. It therefore invents a transition that isn't there.

Figure

The dimension punchline: MF m(T) is dimension-blind, but the exact answer is not

The variational viewpoint reframes the approximation as optimal rather than merely crude. Among all factorized, independent-spin trial distributions, mean-field theory selects the one that minimizes the Gibbs–Bogoliubov upper bound on the free energy. Its "error" is precisely the correlation entropy that a product state cannot represent — which is exactly what matters near a critical point.

The core equations

We start from the Ising Hamiltonian, whose nearest-neighbour coupling is what mean-field theory will approximate.

H=-J\sum_{\langle ij\rangle}S_iS_j-h\sum_i S_i,\qquad S_i=\pm1 \qquad (1)

where: $J>0$ is the ferromagnetic exchange, $\langle ij\rangle$ runs over nearest-neighbour pairs, $h$ is the external field (in energy units), and $S_i$ the Ising spin at site $i$ . Physically, aligned neighbours lower the energy by $J$ ; the field biases spins along $h$ . The mean-field move is to linearize each spin product in its fluctuation about the mean, discarding the term quadratic in fluctuations.

S_iS_j\approx m\,S_i+m\,S_j-m^{2},\qquad m=\langle S_i\rangle \qquad (2)

where: $m$ is the uniform magnetization per spin. Physically, writing $S_i=m+\delta S_i$ and dropping $\delta S_i\,\delta S_j$ is the single approximation — everything else is exact. Inserting the decoupling and counting the $Nz/2$ bonds turns $H$ into a sum of independent single-spin terms plus a constant.

H_{\mathrm{MF}}=\tfrac12 NJz\,m^{2}-(Jzm+h)\sum_i S_i \qquad (3)

where: $z$ is the coordination number and $N$ the number of sites. Physically, each spin now sees only the Weiss field $Jzm+h$ ; the constant $\tfrac12 NJzm^{2}$ corrects for double-counting the bond energy. Computing the single-site thermal average and demanding it equal the assumed $m$ closes the loop — the Weiss equation.

m=\tanh\!\big[\beta(Jzm+h)\big] \qquad (4)

where: $\beta=1/k_BT$ and $h_{\mathrm{eff}}=Jzm+h$ is the effective field. Physically, this is a two-level spin in a self-consistent field; the tanh is the magnetization of a single spin. Expanding for small $m$ at $h=0$ , the nonzero branch appears when the initial slope reaches one, fixing the transition.

k_BT_c=Jz \qquad (5)

where: $T_c$ is the mean-field critical temperature. Physically, for the square lattice $z=4\Rightarrow k_BT_c=4J$ ; for simple cubic $z=6\Rightarrow6J$ ; for the chain $z=2\Rightarrow2J$ — the last being spurious. Expanding the variational free energy to quartic order gives the Landau form whose coefficients yield the exponents.

\beta f=f_0+\tfrac12\beta Jz\,(1-\beta Jz)\,m^{2}+\tfrac1{12}\beta^{4}(Jz)^{4}\,m^{4}-\beta h\,m \qquad (6)

where: $f$ is the free energy per spin and $f_0$ an $m$ -independent reference. Physically, the $m^{2}$ coefficient changes sign at $\beta Jz=1$ (i.e. $T=T_c$ ) while the $m^{4}$ coefficient stays positive — the Landau structure $a(T)m^{2}+b\,m^{4}$ with $a\propto(T-T_c)$ , $b>0$ . Minimizing this Landau free energy delivers the classical exponents and the Curie–Weiss susceptibility.

m\simeq\sqrt3\,\Big(\tfrac{T_c-T}{T_c}\Big)^{1/2},\quad \chi=\frac{1}{k_B(T-T_c)}\propto|t|^{-1},\quad m\propto h^{1/3}\ (T=T_c) \qquad (7)

where: $t=(T-T_c)/T_c$ is the reduced temperature and $\chi$ the isothermal susceptibility. Physically, these are $\beta=\tfrac12$ , $\gamma=1$ , $\delta=3$ , with a specific-heat jump ( $\alpha=0$ ).

Derivation AThe Weiss self-consistency equation from the decoupled Hamiltonian

The goal is to derive the self-consistent magnetization equation by computing the single-site thermal average in the mean-field Hamiltonian and demanding it reproduce the assumed mean. Because $H_{\mathrm{MF}}$ contains no surviving spin–spin coupling, the partition function factorizes site by site, and each spin becomes an elementary two-level system in the Weiss field. We evaluate that single-spin average, recognise it as a hyperbolic tangent of the effective field, and then impose the closure condition that the mean we computed equals the mean we assumed — turning an assumption into an equation whose solutions are the equilibrium states.

Step 1. Factorize the partition function: $Z=e^{-\beta\frac12 NJzm^{2}}\prod_i\sum_{S_i=\pm1}e^{\beta(Jzm+h)S_i}$ , the constant pulling out and each spin summing independently. Why: the decoupling removed all coupling between distinct spins, so the many-body sum splits into $N$ identical one-spin sums.

Step 2. Do the single-site sum: $\sum_{S=\pm1}e^{\beta h_{\mathrm{eff}}S}=2\cosh(\beta h_{\mathrm{eff}})$ with $h_{\mathrm{eff}}=Jzm+h$ . Why: this is the standard two-level partition function, with the Weiss field playing the role of an external field.

Step 3. Compute the average: $\langle S_i\rangle=\dfrac{\sum_S S\,e^{\beta h_{\mathrm{eff}}S}}{\sum_S e^{\beta h_{\mathrm{eff}}S}}=\tanh(\beta h_{\mathrm{eff}})$ . Why: differentiating $\ln(2\cosh)$ or evaluating the two terms directly gives the tanh magnetization of a single spin.

Step 4. Close the loop: identify the computed average with the assumed mean, $m\equiv\langle S_i\rangle=\tanh[\beta(Jzm+h)]$ . Why: consistency is the physical requirement — the field the spins feel must be the field their own magnetization produces. *Verified: verify_aq-sm-mean-field-weiss-theory-ising.py confirms $\langle S\rangle=\tanh(\beta h_{\mathrm{eff}})$ , the decoupling identity $S_iS_j\to m(S_i+S_j)-m^{2}$ dropping exactly $\delta S_i\,\delta S_j$ , and the $H_{\mathrm{MF}}$ coefficients $\tfrac12 NJzm^{2}$ and $-Jzm$ .*

Go deeper · Derivation B · 2 minCritical temperature and the classical exponents from the Landau expansion

Derivation BCritical temperature and the classical exponents from the Landau expansion

The goal is to locate the transition and extract the four mean-field exponents by expanding the Weiss equation and the free energy for small order parameter. We first linearize the self-consistency equation to find where a nonzero solution branches off, giving $k_BT_c=Jz$ . We then expand the variational free energy to quartic order, read off the Landau coefficients, and obtain each exponent by elementary minimization: the order parameter from the field-free minimum, the susceptibility from the curvature above and below, the critical isotherm from the field response at $T_c$ , and the specific heat from the temperature dependence of the minimized free energy. The set turns out dimension-independent and satisfies the standard scaling relations.

Step 1. Linearize at $h=0$ : $m=\tanh(\beta Jz\,m)\approx\beta Jz\,m-\tfrac13(\beta Jz\,m)^{3}$ , so a nonzero root appears when the linear coefficient $\beta_c Jz=1$ , i.e. $k_BT_c=Jz$ . Why: below this temperature the tanh outgrows the line $y=m$ near the origin, producing the symmetric ordered solutions.

Step 2. Order parameter: with $a\propto t$ and $b>0$ , minimizing $f=a m^{2}+b m^{4}-hm$ at $h=0$ gives $m^{2}=-a/(2b)\propto-t$ , so $m\propto|t|^{1/2}$ and $\beta=\tfrac12$ ; the amplitude works out to $m\simeq\sqrt3\,|t|^{1/2}$ . Why: only $t<0$ (below $T_c$ ) yields a real nonzero minimum, growing as the square root.

Step 3. Susceptibility: $\chi^{-1}=\partial h/\partial m=2a+12bm^{2}$ . Above $T_c$ ( $m=0$ ): $\chi^{-1}=2a\propto t$ ; below: $m^{2}=-a/2b$ gives $\chi^{-1}=-4a\propto|t|$ . Both sides give $\gamma=1$ , with the Curie–Weiss form $\chi=1/[k_B(T-T_c)]$ above $T_c$ . Why: the curvature of the free energy at its minimum sets the linear response.

Step 4. Critical isotherm and specific heat: at $t=0$ , $h=4bm^{3}$ so $m\propto h^{1/3}$ and $\delta=3$ ; the minimized free energy has a discontinuous second derivative, giving a specific-heat jump, $\alpha=0$ . The set $(\alpha,\beta,\gamma,\delta)=(0,\tfrac12,1,3)$ satisfies Rushbrooke $\alpha+2\beta+\gamma=2$ and Widom $\gamma=\beta(\delta-1)$ . Why: on the critical isotherm the quadratic term vanishes and the cubic response survives; the finite jump reflects the abrupt onset of order. *Verified: verify_aq-sm-mean-field-weiss-theory-ising.py checks the unit-slope condition $k_BT_c=Jz$ , the Landau coefficients (m² $\propto\beta Jz(1-\beta Jz)$ sign-changing at $T_c$ , m⁴ $>0$ ), the exponents $\beta=\tfrac12,\gamma=1,\delta=3,\alpha=0$ , both scaling relations, the $\sqrt3$ amplitude numerically as $T\to T_c^-$ , and the Curie–Weiss $\chi$ .*

Go deeper · Derivation C · 1 minMean-field theory as a variational upper bound (Gibbs–Bogoliubov)

Derivation CMean-field theory as a variational upper bound (Gibbs–Bogoliubov)

The goal is to show that the Weiss equation is not just a plausible decoupling but the optimal single-site approximation, and that the resulting free energy is a rigorous upper bound on the true one. We invoke the Gibbs–Bogoliubov inequality, which for any trial Hamiltonian bounds the exact free energy from above by the trial free energy plus the average of the difference in the trial ensemble. Choosing a single-site trial field and minimizing over its strength reproduces exactly the Weiss molecular field, proving that mean-field theory is the best product-state description and that it systematically overestimates the free energy — the missing piece being the correlation entropy.

Step 1. Trial Hamiltonian: take independent spins in a variational field, $H_0=-\lambda\sum_i S_i$ , with $\lambda$ to be optimized. Why: a factorized ensemble is the simplest that can capture net magnetization while remaining exactly solvable.

Step 2. Trial averages: $\langle S_i\rangle_0=\tanh(\beta\lambda)\equiv m$ and $F_0=-Nk_BT\ln[2\cosh(\beta\lambda)]$ . Why: the product ensemble makes each spin an independent two-level system in the field $\lambda$ .

Step 3. Bogoliubov bound: $F\le F_0+\langle H-H_0\rangle_0$ with $\langle S_iS_j\rangle_0=m^{2}$ (uncorrelated trial spins), giving $\langle H-H_0\rangle_0=-J\tfrac{Nz}{2}m^{2}+\lambda Nm$ . Why: the inequality follows from convexity of the exponential (Jensen), holding for any trial $H_0$ .

Step 4. Minimize over $\lambda$ : $\partial F_{\mathrm{var}}/\partial\lambda=0\Rightarrow\lambda=Jzm$ — the optimal variational field is the Weiss molecular field, and substituting gives $m=\tanh(\beta Jzm)$ . Hence $F_{\mathrm{exact}}\le F_{\mathrm{MF}}$ . Why: the best product state chooses its internal field to be the mean field, closing the same self-consistency equation from a different route. *Verified: verify_aq-sm-mean-field-weiss-theory-ising.py confirms the decoupling and the identical $T_c$ and exponents; the inequality direction $F_{\mathrm{exact}}\le F_{\mathrm{MF}}$ is the Gibbs–Bogoliubov bound (Sethna, Kardar).*

Limiting cases

Mean-field accuracy is governed by dimensionality and coordination; it becomes exact in well-defined limits and fails badly in others.

Case	Result	Meaning
Infinite-range / Curie–Weiss	$J_{ij}=J/N\ \forall\,i,j$	Every spin couples equally to all others; fluctuations are suppressed as $1/N$ and mean-field theory is exact at all $T$ .
High dimension $d\ge4$	$d\ge d_c=4$	Above the upper critical dimension the Ginzburg criterion holds; mean-field exponents are exact (with logarithmic corrections at $d=4$ ).
One dimension	$k_BT_c^{\mathrm{MF}}=2J$ (spurious)	Mean-field predicts finite- $T$ order, but the exact 1D Ising chain has $T_c=0$ . A qualitative failure driven by unchecked fluctuations.
Two dimensions	$k_BT_c^{\mathrm{MF}}=4J$ vs $2J/\ln(1+\sqrt2)\approx2.269\,J$	Mean-field overestimates $T_c$ by a factor $\approx1.76$ and gives $\beta=\tfrac12$ instead of the exact $\tfrac18$ .

Worked examples

Example 1 — spontaneous magnetization on the square lattice. For the 2D square-lattice Ising model ( $z=4$ ) in mean-field theory, find $T_c$ and compute the magnetization at $T=0.9\,T_c$ .

Step 1. With $z=4$ , the mean-field critical temperature is $k_BT_c=Jz=\mathbf{4J}$ . At $T=0.9\,T_c$ the reduced temperature is $t=(T-T_c)/T_c=-0.1$ , so we expect a nonzero $m$ .

Step 2. Solve $m=\tanh(\beta Jz\,m)=\tanh\!\big(\tfrac{T_c}{T}m\big)=\tanh(1.1111\,m)$ numerically. The nonzero root is $m=\mathbf{0.5254}$ .

Step 3. The leading Landau estimate $m\simeq\sqrt3\,|t|^{1/2}=\sqrt{3\times0.1}=\mathbf{0.5477}$ slightly overshoots the exact self-consistent value $0.5254$ because the full tanh curves below its cubic truncation. Both confirm $\beta=\tfrac12$ : the magnetization rises as the square root of $T_c-T$ .

Example 2 — how badly mean-field overestimates the 2D transition. Compare the mean-field and exact critical temperatures for the square lattice, and quantify the error.

Step 1. Mean-field gives $k_BT_c^{\mathrm{MF}}=4J$ . Onsager's exact solution gives $k_BT_c=2J/\ln(1+\sqrt2)$ . Since $\ln(1+\sqrt2)=0.88137$ , the exact value is $k_BT_c=2J/0.88137=\mathbf{2.2692\,J}$ .

Step 2. The ratio is $T_c^{\mathrm{MF}}/T_c^{\mathrm{exact}}=4/2.2692=\mathbf{1.763}$ : mean-field overestimates $T_c$ by about 76%.

Step 3. The physical reason is that mean-field ignores the order-destroying fluctuations that become ever more important as dimension drops. The same neglect that inflates $T_c$ in 2D turns catastrophic in 1D, where it predicts $k_BT_c=2J$ instead of the correct $T_c=0$ .

Origins · 1 minweiss molecular field and how this was discovered

weiss molecular field

introduced by Pierre Weiss, 1907 · The molecular (or mean) field is the internal effective field that a magnetic material exerts on each of its own spins, proportional to the average magnetization:

h_{\mathrm{mol}}=\lambda M

. Pierre Weiss introduced it in 1907 to explain ferromagnetism, decades before the Ising model was formulated and long before any microscopic understanding of exchange existed. The idea was audacious: a spin does not feel its neighbours individually but the smoothed-out average of the whole magnetized medium, a field the spin itself helps sustain. Applied to the Ising Hamiltonian, the molecular field becomes

Jzm

and yields the self-consistency equation

m=\tanh(\beta Jzm)

, the Curie–Weiss law

\chi=C/(T-T_c)

for the paramagnetic susceptibility, and the classical critical exponents. Weiss's phenomenological field was later understood as the mean-field limit of the quantum exchange interaction, and it remains the prototype of every self-consistent-field method in physics — from Hartree–Fock to BCS theory.

History

1895

— Pierre Curie measures the ferromagnetic transition temperature and Curie's law

1907

— Weiss introduces the molecular (mean) field, explaining ferromagnetism and the Curie–Weiss law

1920

— Lenz proposes the spin model later solved by his student Ising

1925

— Ising solves the 1D chain and finds no transition, wrongly concluding none exists in any dimension

1937

— Landau formulates the general theory of continuous transitions from a symmetric free energy

1944

— Onsager's exact 2D solution shows mean-field exponents are quantitatively wrong (β=1/8, not 1/2)

1966

— Kadanoff's block-spin picture begins to explain why fluctuations invalidate mean field below d=4

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 12 (mean-field theory) · Kardar, Statistical Physics of Fields, Ch. 1–3 · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 7 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 8 · Landau & Lifshitz, Statistical Physics Part 1, §§142–148.

Provenance. The Weiss self-consistency equation $m=\tanh[\beta(Jzm+h)]$ , the critical temperature $k_BT_c=Jz$ , and the classical exponents $\beta=\tfrac12,\gamma=1,\delta=3,\alpha=0$ appear in Pathria & Beale Ch. 12, Kardar (Fields) Ch. 2–3, Reif Ch. 7, Sethna Ch. 8, and Landau–Lifshitz Part 1 §§142–148, and are recomputed in the verify script. The Onsager value $k_BT_c=2J/\ln(1+\sqrt2)\approx2.269\,J$ is the exact 2D square-lattice result; the mean-field/exact ratio $4/2.2692\approx1.763$ is verified. The Gibbs–Bogoliubov variational route and the resulting upper bound $F_{\mathrm{exact}}\le F_{\mathrm{MF}}$ are standard. No nonstandard numbers are introduced.

Builds on: The Ising Model: Definition and Hamiltonian · Order Parameter and Spontaneous Magnetization. Leads to: Bethe-Peierls Approximation · Onsager Solution of the Two-Dimensional Ising Model · Binary Alloy Order-Disorder Transition · Peierls Argument for Long-Range Order.

What is the difference between "mean-field theory" and "Weiss molecular-field theory"?

They are the same physics arrived at from two directions. Weiss (1907) postulated phenomenologically that each spin feels an internal field proportional to the magnetization, $h_{\mathrm{mol}}=\lambda M$ , to explain ferromagnetism — long before the Ising model existed. Mean-field theory is the modern derivation: linearize the spin–spin interaction in fluctuations, or variationally optimize a product trial state, and the effective field that emerges is exactly the Weiss molecular field $Jzm$ . So "Weiss theory" names the historical molecular-field idea, and "mean-field theory" names the systematic approximation that reproduces and generalizes it. Both give $m=\tanh(\beta Jzm)$ and $k_BT_c=Jz$ .

Why does the critical temperature depend on the coordination number

z

but not the dimension directly?

In mean-field theory a spin only "knows" its number of neighbours $z$ , because the neighbour sum is replaced by $z\langle S\rangle$ . Dimension enters only through $z$ (e.g. $z=2,4,6$ for the chain, square, and cubic lattices), so $k_BT_c=Jz$ scales with connectivity. This is precisely the theory's blind spot: the geometry of how those neighbours are arranged — which controls fluctuations — is invisible to it. That is why mean-field theory gives the same exponents in every dimension and gets $T_c$ increasingly wrong as $d$ drops, culminating in the spurious 1D transition.

How can I see that mean-field theory becomes exact for infinite-range interactions?

Take $J_{ij}=J/N$ equal between every pair of spins (the Curie–Weiss model). Then the interaction energy is $-\tfrac{J}{2N}(\sum_i S_i)^2=-\tfrac{JN}{2}m^2$ , a function of the total magnetization alone, with no residual correlations. The fluctuations of $m$ about its mean scale as $1/\sqrt N$ and vanish in the thermodynamic limit, so replacing $m$ by its average is exact, not approximate. A saddle-point (Hubbard–Stratonovich) evaluation makes this rigorous: the mean-field free energy is the exact large- $N$ limit, which is why mean-field theory is also called the "infinite-dimensional" or "infinite-coordination" limit.

If mean-field exponents are wrong below

d=4

, why is the theory still worth learning?

Because it gets the structure right even when the numbers are wrong, and it is exact where it counts. Mean-field theory correctly identifies that there is a continuous transition, an order parameter, a diverging susceptibility, and a broken symmetry; it gives a concrete $T_c$ estimate and a first picture of the ordered phase. It is exact above $d_c=4$ and for long-range or infinite-range couplings, and it is the zeroth-order term of the $\epsilon$ -expansion — you cannot compute the corrections without first having the mean-field answer. Its failures are also pedagogically precious: the spurious 1D transition is the cleanest motivation for studying fluctuations and the renormalization group.

Where exactly does the factor

\tfrac12

in the constant term

\tfrac12 NJzm^2

come from?

It is a double-counting correction. There are $Nz/2$ bonds in a lattice with $N$ sites of coordination $z$ (each bond shared by two sites, hence the $/2$ ). When you decouple $-J\sum_{\langle ij\rangle}[m(S_i+S_j)-m^2]$ , the $-m^2$ per bond contributes $-J\cdot(Nz/2)\cdot(-m^2)=+\tfrac12 NJzm^2$ . Without this constant the free energy would be wrong by a term $\propto m^2$ , spoiling the Landau expansion and the location of $T_c$ . It is exactly the piece that prevents you from counting each spin's mean-field energy twice, once for each end of every bond.

Does mean-field theory give the right answer for the antiferromagnet too?

Yes, with a small modification: use two sublattices. For $J<0$ (antiferromagnetic) neighbouring spins prefer to anti-align, so you introduce sublattice magnetizations $m_A,m_B$ and a self-consistency pair $m_A=\tanh[\beta(|J|z\,m_B+h)]$ and vice versa. The staggered magnetization $m_A-m_B$ becomes the order parameter, and the Néel temperature plays the role of $T_c$ , again $k_B T_N=|J|z$ in mean field. The same over-estimation and dimension-blindness apply. This two-sublattice mean-field theory is the workhorse for Néel order and generalizes directly to the binary-alloy order–disorder problem, where the two sublattices are the two atomic species.