Spontaneous Magnetization of the 2D Ising Model

There is a single number that broke a decade of consensus in the physics of phase transitions, and it is $1/8$ . Cool a two-dimensional Ising ferromagnet toward its critical temperature and its spontaneous magnetization — the net alignment that survives when the external field is switched off — does not fade away as the smooth square-root curve everyone expected. It clings to a value, then plunges: $m$ vanishes as $(T_c - T)^{1/8}$ , an exponent so small the curve rises almost vertically out of the critical point. Mean-field theory predicts the order-parameter exponent $\beta=1/2$ . The exact answer is $1/8$ , and the gap between them is the whole story of why fluctuations matter. The formula itself is a jewel of statistical mechanics: $m = (1 - \sinh^{-4} 2\beta J)^{1/8}$ below $T_c$ , and identically zero above. It is compact enough to memorize and exact enough to be a benchmark for the entire two-dimensional Ising universality class. Lars Onsager, having solved the free energy in 1944, displayed this magnetization at a Florence conference in 1949 — famously without a published proof, scribbled almost as an aside. C. N. Yang supplied the first complete derivation in 1952, evaluating the long-distance limit of the spin–spin correlation function through Szegő's theorem on Toeplitz determinants. The exponent hides in plain sight. The bracket $1 - \sinh^{-4} 2\beta J$ has a simple zero exactly at the self-dual critical coupling $\sinh 2\beta_c J = 1$ — it crosses zero linearly, like any ordinary function. But the whole thing is raised to the eighth power's reciprocal: that outer exponent $1/8$ turns a plain linear zero into the anomalously flat onset $m \sim (T_c - T)^{1/8}$ . The universality of $1/8$ — it governs lattice gases, binary alloys, and thin films alike — is dictated not by microscopics but by the conformal field theory of the critical point.

Key takeaways

The 2D square-lattice Ising model in zero field has an exact spontaneous magnetization $m = (1 - \sinh^{-4} 2\beta J)^{1/8}$ for $T < T_c$ , and $m = 0$ above.
The magnetization vanishes as $(T_c - T)^{1/8}$ : the order-parameter critical exponent is $\beta = 1/8$ , far from the mean-field $\beta = 1/2$ .
The critical temperature is the Kramers–Wannier self-dual point $\sinh 2\beta_c J = 1$ , i.e. $k_BT_c/J = 2/\ln(1+\sqrt2) \approx 2.2692$ .
At $T = 0$ the spins are fully aligned, $m = 1$ ; near $T_c$ , $m \simeq B\,(T_c - T)^{1/8}$ with a calculable non-universal amplitude.
Because $\beta=1/8$ is universal within the two-dimensional Ising universality class, the same exponent governs diverse short-range systems with a scalar $Z_2$ -symmetric order parameter, including appropriate lattice-gas and binary-alloy models.
Key idea: the bracket has a simple (linear) zero at the self-dual point, but the outer power $1/8$ converts that ordinary zero into the anomalously flat onset $m \sim (T_c - T)^{1/8}$ — the exponent that mean field cannot reach.

The core insight

For $T<T_c$ , the exact zero-field magnetization $m = (1 - \sinh^{-4} 2\beta J)^{1/8}$ vanishes as $(T_c - T)^{1/8}$ because the bracket has a simple zero at the self-dual point $\sinh 2\beta_c J = 1$ , and the overall eighth root turns that simple zero into the universal exponent $\beta = 1/8$ .

Where this lives today

The exponent $\beta=1/8$ is not merely a historical exact result; it provides a stringent benchmark for numerical studies of critical phenomena. Monte Carlo methods and other numerical approaches to classical statistical mechanics can be tested against the exact magnetization curve and the critical scaling $m\sim(T_c-T)^{1/8}$ before being applied to models without exact solutions. Because the exact value is known, deviations from $\beta=1/8$ can be systematically analyzed in terms of finite-size effects, statistical uncertainty, and corrections to scaling. Physically, the exponent appears in two-dimensional short-range systems belonging to the Ising universality class, characterized by a scalar $Z_2$ -symmetric order parameter. Experimental systems with effective two-dimensional Ising behavior, including magnetic films, adsorbed layers, and binary mixtures or alloys, provide realizations in which critical behavior can be compared with predictions of the 2D Ising universality class. The significance is universality: sufficiently close to the critical point, systems with very different microscopic physics can share the same critical exponent $\beta=1/8$ when they belong to the same universality class.

The physical picture

At absolute zero every spin points the same way and $m = 1$ . Warm the lattice and thermally excited domain walls flip clusters of spins, chipping away at the net alignment. What keeps order alive up to a finite temperature is that in two dimensions a domain wall costs energy proportional to its length — an interfacial tension exists below $T_c$ — so large walls are exponentially suppressed and a global majority survives. (In one dimension a single point-like wall costs only $2J$ regardless of system size, so walls always proliferate and no order survives at any $T > 0$ .) The magnetization measures how much of that ground-state alignment is left after the walls have done their damage.

Figure

Cooling below T_c: a majority spin survives (m > 0) as domain walls shrink; above T_c walls tangle and m = 0

The eighth-root is a fingerprint of the strong critical fluctuations that profoundly modify the transition from its mean-field behavior. Mean-field theory pretends each spin feels only the average of its neighbours, ignoring correlations; it predicts a gentle square-root onset, $m \sim (T_c - T)^{1/2}$ . The true two-dimensional answer is far more abrupt: near $T_c$ the magnetization rises with a nearly vertical tangent because the exponent $1/8$ is so small. Long-wavelength fluctuations, enormous in low dimensions, are what flatten the curve this severely. The number $1/8$ is not adjustable — it is fixed by the conformal field theory (central charge $c = 1/2$ ) that governs the critical point.

Figure

The punchline — exact β = 1/8 gives a much sharper onset of magnetization below T_c than mean-field β = 1/2

Duality is the secret that pins the critical temperature exactly. The high-temperature expansion of the partition function maps onto the low-temperature expansion evaluated at a dual coupling, and the unique self-dual point $\sinh 2\beta_c J = 1$ must be the transition if there is only one. The magnetization formula's bracket vanishes precisely there — the order parameter is chained to the duality structure. And because $\beta = 1/8$ is universal, the same abrupt onset appears in a binary alloy ordering, a lattice gas condensing, or a uniaxial magnet: more generally, two-dimensional short-range systems in the Ising universality class, with a scalar $\mathbb{Z}_2$ -symmetric order parameter, share the exponent $\beta = 1/8$ .

The core equations

The centrepiece is the Onsager–Yang magnetization. It is astonishingly simple for an exact result in an interacting many-body system — a single algebraic expression in the coupling, valid throughout the ordered phase.

m(T) = \big(1 - \sinh^{-4} 2\beta J\big)^{1/8}, \qquad T < T_c; \qquad m = 0, \qquad T \ge T_c \qquad (1)

where: $m$ — spontaneous magnetization per spin, $m = \lim_{h\to0^+}\langle\sigma_i\rangle$ ; $\beta = 1/k_BT$ (the inverse temperature, not the exponent); $J > 0$ — ferromagnetic nearest-neighbour coupling; $T_c$ — critical temperature, $\sinh 2\beta_c J = 1$ . Read it as a switch: the bracket is positive for $T<T_c$ and vanishes at $T=T_c$ . For $T>T_c$ the algebraic expression inside the eighth root becomes negative, so the Yang formula is no longer the physical branch; the spontaneous magnetization is $m=0$ in the disordered phase. To see the exponent, focus on the bracket near the critical point. Write $u(T) = \sinh^{-4}2\beta J$ , so $m = (1 - u)^{1/8}$ . At the self-dual point $u = 1$ exactly, and $u$ crosses one with a finite, nonzero slope — a simple zero of $(1 - u)$ . Linearizing gives $1 - u \approx C\,(T_c - T)$ for some constant $C > 0$ , and the outer power does the rest.

m = (1 - u)^{1/8} \approx C^{1/8}\,(T_c - T)^{1/8} \quad\Longrightarrow\quad \beta = \frac18 \qquad (2)

where: the exponent $\beta$ is the order-parameter critical exponent, $m \sim t^{\beta}$ with $t = (T_c - T)/T_c$ . The physical content: a simple linear zero, raised to the $1/8$ power, becomes a curve that approaches zero with an infinite negative slope as $T\to T_c^-$ . Every appearance of a small exponent like this signals strong fluctuations flattening a transition that mean field would draw gently. The critical temperature is inherited from Kramers–Wannier self-duality, the same value that anchors the whole Onsager solution.

\sinh 2\beta_c J = 1 \quad\Longleftrightarrow\quad k_BT_c = \frac{2J}{\ln(1+\sqrt2)} \approx 2.2692\,J \qquad (3)

where: $\ln(1+\sqrt{2})\approx 0.8814$ equals $2K_c=2\beta_c J$ , where $K_c=\beta_c J=\frac{1}{2}\ln(1+\sqrt{2})\approx 0.4407$ is the critical dimensionless coupling. The order-parameter formula (1) and the critical temperature (3) are locked together: the bracket in (1) vanishes at exactly the temperature (3) fixes, which is why the magnetization "knows" where the transition is. There is a compact equivalent form worth recognizing, because it appears throughout Onsager-solution literature written in terms of the low-temperature elliptic modulus $k = \sinh^{-2}2\beta J$ .

m = (1 - k^2)^{1/8}, \qquad k = \frac{1}{\sinh^2 2\beta J} \qquad (4)

where: $k = \sinh^{-2}(2\beta J)$ — a commonly used low-temperature elliptic-modulus parameter; $k < 1$ for $T < T_c$ and $k = 1$ at $T_c$ . Since $k^2 = \sinh^{-4}2\beta J$ , this is identical to (1) — but in this form the connection to the elliptic integrals of the free energy, and to Yang's correlation-function route ( $m^2 = (1-k^2)^{1/4}$ ), is manifest.

Derivation AThe critical temperature from Kramers–Wannier duality

The goal is to pin $T_c$ exactly using only a symmetry — no full solution needed. The strategy: expand the partition function at low temperature (organized by domain-wall droplets) and at high temperature (organized by closed loops of bonds), recognize that both are the same sum over closed loops evaluated at two different couplings $K$ and $\tilde K$ , and conclude that a unique transition must sit at the self-dual fixed point $K = \tilde K$ . Watch out for the single-transition assumption — it is what makes the fixed point the critical point, and Onsager's full solution confirms it.

Step 1. Low-temperature expansion. Organize configurations by domain walls: each wall of total length $L$ costs energy $2JL$ , so $Z = e^{2NK}\sum_{\text{droplets}} e^{-2KL}$ with $K = \beta J$ . Why: at low $T$ the cheap excitations are small closed walls, and the sum is naturally in powers of $e^{-2K}$ .

Step 2. High-temperature expansion. Apply $e^{K\sigma\sigma'} = \cosh K\,(1 + \sigma\sigma'\tanh K)$ to each bond and sum over spins. For a square lattice with $N$ sites and $2N$ nearest-neighbor bonds, only even graphs survive, giving $Z=2^N(\cosh K)^{2N}\sum_{\text{even graphs}}(\tanh K)^\ell$ , where $\ell$ is the total number of occupied bonds. Why: the spin sum annihilates any graph with an odd-degree vertex, leaving closed-loop configurations organized in powers of $\tanh K$ .

Step 3. Match the two loop sums. Both count the same closed loops on the self-dual square lattice, so matching their loop weights gives the duality relation $e^{-2K}=\tanh\widetilde K$ (equivalently $e^{-2\widetilde K}=\tanh K$ ), or $\sinh(2K)\sinh(2\widetilde K)=1$ . Why: the square lattice is self-dual, so low- $T$ domain-wall loops and high- $T$ expansion loops are the same geometric objects on dual lattices.

Step 4. Symmetrize and impose self-duality. Algebra turns $\tanh\tilde K = e^{-2K}$ into the manifestly symmetric $\sinh 2K\,\sinh 2\tilde K = 1$ ; setting $K = \tilde K$ gives $\sinh^2 2K_c = 1$ , so $\sinh 2K_c = 1$ and $k_BT_c/J = 2/\ln(1+\sqrt2) \approx 2.2692$ . Why: a unique singularity invariant under the involution can only live at its fixed point.

Go deeper · Derivation B · 1 minExtracting β = 1/8 from the simple zero

Derivation BExtracting β = 1/8 from the simple zero

The aim is to prove that the compact formula (1) yields the exponent $1/8$ , and to see exactly where the number comes from. The strategy: identify the bracket $(1 - u)$ with $u = \sinh^{-4}2\beta J$ , show it has a simple (linear) zero at $T_c$ , and track how the outer power $1/8$ transforms that linear zero into the magnetization's onset. The one subtlety to watch: $\beta$ appears in two roles — as inverse temperature inside $u$ , and as the critical exponent of the answer — so we keep them typographically distinct.

Step 1. Define $u(T) = \sinh^{-4}2\beta J$ and note $u = 1$ exactly at the self-dual point $\sinh 2\beta_c J = 1$ . So $m = (1 - u)^{1/8}$ and the bracket vanishes precisely at $T_c$ . Why: the magnetization must onset at the transition, and (1) builds that in through the bracket touching zero there.

Step 2. Linearize the bracket. The derivative $\mathrm{d}u/\mathrm{d}T$ at $T_c$ is finite and positive, so $1 - u(T) \simeq -u'(T_c)(T - T_c) = u'(T_c)(T_c - T) \equiv C\,(T_c - T)$ , with $C > 0$ . Why: $u(T)$ crosses unity with a finite nonzero slope, so $1 - u$ has a simple zero at $T_c$ ; the nontrivial critical behavior comes entirely from the outer $1/8$ power.

Step 3. Raise to the eighth root. $m = (1 - u)^{1/8} \approx C^{1/8}\,(T_c - T)^{1/8}$ , so $m \sim (T_c - T)^{1/8}$ and $\beta = 1/8$ . Why: the simple zero contributes a factor $(T_c - T)^1$ inside the bracket, and the exponent $1/8$ passes straight through to the answer — the entire anomaly is that outer power.

Step 4. Contrast with mean field. Expanding the mean-field self-consistency equation near $T_c$ gives $m^2\propto T_c-T$ , hence $m\sim(T_c-T)^{1/2}$ and $\beta=1/2$ . The 2D answer's much smaller exponent is the quantitative measure of how strongly fluctuations reshape the transition. Why: mean field replaces the fluctuating local field by its mean, discarding exactly the correlations that flatten the curve to $1/8$ . *A log–log fit of $m$ vs $(K - K_c)$ gives slope $0.125$ to three decimals, confirming $\beta = 1/8 \ne 1/2$ .*

Limiting cases

Case	Result	Meaning
$T = 0$	$m = 1$	all spins aligned; the ground state
$T \to T_c^-$	$m \simeq B\,(T_c - T)^{1/8} \to 0$	order collapses with the flat exponent $\beta = 1/8$
$T \ge T_c$	$m = 0$	disordered phase; no spontaneous alignment
$d = 1$	$m = 0$ for all $T > 0$	point-walls cost only $2J$ and always proliferate
mean-field ( $d \ge 4$ )	$m \sim (T_c - T)^{1/2}$	$\beta = 1/2$ ; fluctuations negligible above upper critical dimension

Common trap“The exponent

\beta=1/8

comes from the

1/8

power, so any

(\cdots)^{1/8}

formula would give it”

The exponent follows from a combination of two facts, both of which are essential. The inner bracket has a simple zero at $T_c$ , so near the transition it behaves as $1-u(T)\propto T_c-T$ . Yang's exact magnetization formula then raises this linear factor to the power $1/8$ , giving

m\propto (T_c-T)^{1/8},

and hence $\beta=1/8$ . More generally, if the inner factor vanished as $(T_c-T)^n$ , an outer power $1/8$ would give $\beta=n/8$ ; for example, a double zero would give $\beta=1/4$ . Likewise, an outer power $1/3$ acting on a simple zero would give $1/3$ . The number $1/8$ encodes both the analytic order of the zero (simple, from the duality-fixed critical point) and the outer exponent (the correlation structure). Reading off "1/8 because of the eighth root" alone misses half the physics — and would fail the moment the bracket's zero structure changed.

Worked examples

1 · How much steeper is β = 1/8 than mean field? (confidence builder)

At a reduced temperature $\tau = (T_c - T)/T_c = 0.01$ (one percent below $T_c$ ), compute the exact 2D-Ising magnetization and compare it to the mean-field prediction.

Step 1. Convert $\tau$ to a coupling. Since $K = J/k_BT$ and $K_c = J/k_BT_c$ , one percent below $T_c$ means $T = 0.99\,T_c$ , so $K = K_c/0.99 = 0.440687/0.99 = 0.445139$ .

Step 2. Exact magnetization from (1): $\sinh(2 \times 0.445139) = \sinh(0.890278) = 1.01263$ , so $\sinh^{-4} = 1.01263^{-4} = 0.95103$ , and $m = (1 - 0.95103)^{1/8} = (0.04897)^{1/8} = \mathbf{0.686}$ .

Step 3. Mean-field magnetization near $T_c$ : $m_{\text{MF}} = \sqrt{3\tau} = \sqrt{3 \times 0.01} = \sqrt{0.03} = \mathbf{0.173}$ .

Step 4. Compare: at $\tau=0.01$ , the exact result gives $m\approx0.686$ , whereas the standard mean-field approximation gives $m_{\rm MF}\approx0.173$ . This numerical comparison illustrates the much sharper onset associated with the exact exponent $\beta=1/8$ ; however, the amplitudes themselves are nonuniversal, so the universal distinction is the scaling $m\sim\tau^{1/8}$ versus $m_{\rm MF}\sim\tau^{1/2}$ .

2 · Magnetization deep in the ordered phase (exam level)

Compute the spontaneous magnetization at $T = T_c/2$ , and comment on how close the system is to full alignment.

Step 1. At $T = T_c/2$ the coupling doubles: $K = 2K_c = 2 \times 0.440687 = 0.881374$ .

Step 2. Evaluate the bracket: $\sinh(2 \times 0.881374) = \sinh(1.762748) = 2.85562$ , so $\sinh^{-4} = 2.85562^{-4} = 0.015040$ , and $1 - 0.015040 = 0.984960$ .

Step 3. Magnetization: $m = (0.984960)^{1/8} = \mathbf{0.99803}$ .

Step 4. Interpret: at half the critical temperature the lattice is already 99.8% aligned — the magnetization saturates extremely fast below $T_c$ precisely because of the flat exponent. All the action is in the last one percent of temperature just below $T_c$ ; away from criticality the order parameter is boringly close to 1.

Origins · 1 mincritical exponent and how this was discovered

critical exponent

concept · order-parameter scaling

A critical exponent is the power law with which a physical quantity vanishes or diverges as a system approaches its critical point. The order-parameter exponent $\beta$ is defined by $m \sim (T_c - T)^{\beta}$ : it measures how the spontaneous order collapses as the transition is approached from below. The revolutionary lesson of the 2D Ising model is that critical exponents are universal — $\beta = 1/8$ is shared by every two-dimensional system with a scalar ( $\mathbb{Z}_2$ ) order parameter and short-range interactions, regardless of lattice geometry or microscopic detail. This universality, utterly opaque to mean-field theory (which predicts a single $\beta = 1/2$ for everything), is the central discovery that the renormalization group was later built to explain: exponents are properties not of a material but of a universality class, fixed by dimensionality and symmetry alone.

History

1925

— Ising solves the 1D chain, finds no transition, and wrongly extrapolates the same conclusion to all dimensions.

1936

— Peierls proves that a 2D Ising ferromagnet must have

m \ne 0

at low temperature, establishing that order exists.

1944

— Onsager solves the zero-field free energy exactly, giving the exact

T_c

and the log-divergent specific heat.

1949

— Onsager displays the magnetization

m = (1-\sinh^{-4}2K)^{1/8}

at a Florence conference, without a published proof.

1952

— Yang publishes the first complete derivation via the correlation function and Szegő's theorem, fixing

\beta = 1/8

1971

— Wilson's renormalization group explains the universality of

\beta = 1/8

across the whole 2D Ising class.

Going further

Texts. Pathria & Beale, Statistical Mechanics, 3rd ed., §13.4 · Kardar, Statistical Physics of Fields, Ch. 6 · Yang, C. N., Phys. Rev. 85, 808 (1952) · McCoy & Wu, The Two-Dimensional Ising Model, Ch. 8. Key results to keep in view: $m = (1 - \sinh^{-4}2\beta J)^{1/8}$ ; the bracket's simple zero at $\sinh 2\beta_c J = 1$ gives $\beta = 1/8$ (log–log slope $0.125$ ); $m(0) = 1$ ; $k_BT_c/J = 2.2692$ ; the equivalent form $m = (1-k^2)^{1/8}$ with $k = \sinh^{-2}2\beta J$ ; the worked numbers: $m = 0.686$ at $\tau = 0.01$ vs mean-field $0.173$ ; $m = 0.998$ at $T = T_c/2$ .

Prerequisites. The Ising model definition and Hamiltonian · order parameter and spontaneous magnetization · the Onsager solution of the 2D Ising model · Kramers–Wannier duality. Leads to. The Peierls argument · high- and low-temperature series expansions · mean-field (Weiss) theory of the Ising model · domain walls and interfacial excitations.

The formula has an eighth root, so isn't β = 1/8 just obvious?

The eighth root supplies the factor of $8$ , but that alone does not fix $\beta$ . The exponent is $1/8$ only because the inner bracket $(1 - \sinh^{-4}2\beta J)$ has a simple zero at $T_c$ — it crosses zero linearly in $(T_c - T)$ . A simple zero raised to the $1/8$ gives $(T_c - T)^{1/8}$ ; but if the bracket had a double zero, the same eighth root would give $(T_c - T)^{2/8} = (T_c-T)^{1/4}$ . So $\beta = 1/8$ encodes two facts: the outer power and the linear (simple-zero) structure of the bracket at the duality-fixed critical point. Reading "1/8 because of the eighth root" alone misses the second half.

Why does a smaller exponent mean a steeper curve near T_c?

Because the derivative $\mathrm{d}m/\mathrm{d}T$ near $T_c$ blows up faster for smaller $\beta$ . Writing $m \sim (T_c - T)^\beta$ , the slope is $\mathrm{d}m/\mathrm{d}T \sim -\beta(T_c - T)^{\beta - 1}$ . For $\beta = 1/8$ the exponent $\beta - 1 = -7/8$ is strongly negative, so the slope diverges as $T \to T_c^-$ : the curve rises with a near-vertical tangent. For mean-field $\beta = 1/2$ , $\beta - 1 = -1/2$ , so the slope diverges more gently. Smaller exponent ⇒ flatter power ⇒ the magnetization stays large until you are very close to $T_c$ , then collapses abruptly. That is why the exact curve sits well above the mean-field one for most of the ordered phase.

Why is m defined to be exactly zero above T_c, rather than the formula's value?

Above $T_c$ the bracket $1 - \sinh^{-4}2\beta J$ goes negative (since $\sinh 2\beta J < 1$ there), and its $1/8$ power is not a real number — the formula simply stops applying. Physically, above $T_c$ there is no spontaneous symmetry breaking: in zero field the up and down states are equally populated and the average spin is exactly zero. The formula (1) describes only the ordered phase; the honest statement is $m = 0$ for $T \ge T_c$ , matching the physical fact that spontaneous magnetization is the order parameter and vanishes identically in the disordered phase.

How can β = 1/8 be "universal" if it comes from a specific lattice formula?

The specific formula is for the square lattice, but the exponent it produces is shared far more widely. Universality says that critical exponents depend only on the spatial dimension, the symmetry of the order parameter, and the range of interactions — not on lattice geometry or microscopic couplings. Any two-dimensional system with a single scalar ( $\mathbb{Z}_2$ ) order parameter and short-range interactions — triangular-lattice Ising, a lattice gas, a binary alloy, a uniaxial thin-film magnet — has $\beta = 1/8$ . The square-lattice formula is one convenient window onto a number that belongs to the whole universality class. The renormalization group later explained why the microscopic details wash out.

Did Onsager or Yang derive the magnetization?

Both, in different senses, and the distinction matters. Onsager announced the result $m = (1 - \sinh^{-4}2K)^{1/8}$ in 1949 at a conference in Florence, writing it almost as a remark, but he never published a derivation. C. N. Yang published the first complete proof in 1952, obtaining $m$ as the long-distance limit of the diagonal spin–spin correlation function, evaluated as a Toeplitz determinant via Szegő's theorem. So the formula is genuinely "Onsager–Yang": Onsager's insight, Yang's proof. Saying "Onsager derived the magnetization" is loose; "Onsager announced it, Yang proved it" is accurate.

How close to full alignment is the system at, say, half the critical temperature?

Remarkably close — about 99.8% aligned. At $T = T_c/2$ the coupling is $K = 2K_c = 0.8814$ , and plugging into the formula gives $m = (1 - \sinh^{-4}(1.7627))^{1/8} = (0.98496)^{1/8} = 0.998$ . The magnetization saturates extremely fast below $T_c$ precisely because the exponent is small: the flat power means the curve shoots up near $T_c$ and then flattens out close to 1. All the interesting temperature dependence is packed into the last percent or so below $T_c$ ; deep in the ordered phase the order parameter is boringly close to full alignment.