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Ferromagnetism is collective magnetic order with spontaneous magnetization below a Curie temperature. Interactions favor aligned magnetic moments; the Ising model represents a two-orientation version with positive coupling.

A magnetic domain can possess spontaneous magnetization even when other domains cancel its contribution to the whole specimen. Compare this spatial cancellation between domains with antiferromagnetism, where opposing magnetic sublattices compensate within the ordered structure, and ferrimagnetism, where their contributions are generally unequal. The magnetic sublattice diagram makes the three local patterns explicit.

Smoothly sweeping an applied magnetic field can reveal abrupt magnetization changes through the Barkhausen effect. A pickup coil detects the resulting changes of magnetic flux, making collective magnetic domain wall motion accessible as an electrical signal.

The infrared bound controls the nonzero Fourier modes of a spin correlation by the inverse lattice dispersion. Its momentum integral is finite in dimension at least three and can force spontaneous magnetization at low temperature.

The Ising model is a spin model with an Ising spin ${\sigma }_v\in \{-1,1\}$ at every graph vertex $v$. For a finite graph with vertex set $V$ and edge set $E$, a uniform coupling $J$ and field parameter $h$ give the HamiltonianEach edge is counted once. Here $T>0$ is temperature, $k_B$ is the Boltzmann constant, and $\beta$ is inverse temperature; $J$ and $h$ have units of energy. For magnetic applications, $h$ includes the magnetic moment multiplying the applied magnetic field. The canonical ensemble assigns the Boltzmann distributionThe partition function $Z$ makes the probabilities sum to one. Positive $J$ rewards equal neighboring Ising spins, modeling ferromagnetism; negative $J$ rewards opposite neighbors, giving an antiferromagnetic Ising model. Positive $h$ rewards $+1$, while negative $h$ rewards $-1$.^[ref]

A single local rule can organize a whole graph: rewarding agreement on each edge makes all-equal Ising spin patterns favorable even though no edge directly couples distant vertices. The four-spin Ising ring makes this visible with only sixteen microstates. It also separates three different questions: whether neighbors agree, whether the mean spin magnetization is nonzero, and how large its thermal fluctuations are.

A motivating experiment is the University of Colorado Boulder's Curie-temperature demonstration. An iron rod suspended from a twisting wire on a stand is initially held near a permanent magnet by ferromagnetism. A gas burner supplied from a propane cylinder heats the rod; the readout is its motion away from the magnet, followed by return as it cools. Repeated cooling supplies a reversible comparison on the same sample. The source explicitly calls for heating the rod while keeping the magnet cool, since heating the magnet can reduce its strength. The inference is that warming the sample weakens its magnetic attraction across the Curie temperature. This is a qualitative force readout, with no calibrated temperature or magnetization trace supplied: it motivates competition between alignment and thermal fluctuations, but does not identify $J$, measure individual Ising spins, or realize a four-spin Ising ring.^[ref]

Ernst Ising's 1925 paper examined a linear chain of two-orientation magnetic elements with neighboring interactions. In its calculation for a large chain, the zero-field mean magnetization vanishes and the model has no spontaneous magnetization. The inspected English translation states this immediately after equation (8) in §2. This historical result concerns the chain; it does not establish the behavior of a two-dimensional Ising model.^[ref]

Define the instantaneous spin magnetization $m=\frac{1}{4}{\sum }_i{\sigma }_i$. Global reversal sends $m$ to $-m$ without changing the zero-field energy. Pairing every microstate with its reversed partner proves $⟨m⟩=0$ at every positive temperature, even when almost all probability lies on the two fully aligned patterns. A vanishing mean does not imply weak alignment: the limiting probability distribution is half at $m=+1$ and half at $m=-1$.

The Ising ring magnetization fluctuations measure this distinction through variance. In the $d=0$ class, both patterns have $m^2=1$. Among the $d=2$ patterns, eight have $m^2=1/4$ and four have $m^2=0$; the two $d=4$ patterns also have $m^2=0$. ConsequentlyIt equals $6/17$ at $K=(\ln 2)/4$, approaches $1/4$ for four independent fair Ising spins at $K\to 0$, and approaches $1$ as $K\to \infty$. Unlike the energy variance, the spin magnetization variance remains large in the low-temperature symmetric mixture.

The same thermal fluctuation controls the field response. Allow $h$ to vary before setting it to zero; differentiating the Boltzmann distribution inserts $\beta {\sum }_i{\sigma }_i=4\beta m$, givingThis is a spin-normalized magnetic susceptibility with respect to the energy-valued parameter $h$; conversion to physical magnetization and magnetic field requires the magnetic moment and sample volume factors.

For finite $N$, $Z$ is a finite positive sum of exponential functions. Its free energy $-k_{B}T\log Z$ is a real analytic function for $T>0$, so the smooth heat capacity peak here is not a critical phase transition. Spontaneous magnetization concerns the order of limits: first a thermodynamic limit, then removal of a symmetry-breaking field. The four-spin Ising ring alone establishes neither that limit nor any two-dimensional Ising model result.^[ref]

An order parameter distinguishes phases by a macroscopic quantity that changes at the transition. For an Ising model ferromagnet it is the magnetization per Ising spin, $M=N^{-1}{\sum }_n⟨{\sigma }_n⟩$. At zero magnetic field the statistical Hamiltonian is invariant under spin inversion symmetry. The disordered phase preserves this symmetry, whereas a selected ordered phase has $M\ne 0$ and a spin-reversed partner with $-M$. At finite volume with symmetric boundary conditions the expected value is zero even below the transition; spontaneous magnetization means taking the thermodynamic limit before sending a symmetry-breaking field to zero.

Landau-Ginzburg theory introduces a coarse-grained local order parameter $\varphi (x)$ and a symmetry-constrained free-energy functional. For a short-range system with scalar spin inversion symmetry, a useful expansion isThe coefficients vary smoothly with microscopic control parameters at the level of the Landau approximation; normally $r=A(T-T_c)$ with $A>0$. The gradient term penalizes spatial variation. The magnetic field is conjugate to the order parameter, and odd powers at $h=0$ are excluded by spin inversion symmetry. With $u>0$, the sextic term may be omitted locally; with $u<0$, a stabilizing positive $v$ is essential. The equilibrium uniform order parameter minimizes the local potential $V(M)$, givingThis equation of state follows from minimization, rather than being imposed as an independent assumption. Vector or tensor order parameters and their allowed invariants describe other symmetry classes.

Use ${\beta }_T=1/(k_{B}T)$ for inverse temperature, to distinguish it from the order-parameter critical exponent $\beta$. In a translation-invariant pure thermodynamic phase, the connected two-point correlation function isFor finite correlation length, at $|r|\gg \xi$ it decays exponentially, possibly with an algebraic prefactor; ${\xi }^{-1}$ is the asymptotic exponential decay rate. Below an ordered transition the pure phase is selected by taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$. A zero-field symmetric mixture of the two ordered phases has a nondecaying connected term and does not describe the usual pure-phase magnetic susceptibility.

Let $\mathcal{F}=-k_{B}T\log Z$ denote extensive physical free energy and $S={\sum }_r{\sigma }_r$. Since the field contributes $-hS$ to the statistical Hamiltonian, differentiation givesExpanding the variance and using translation invariance proves the correlation-function susceptibility sum rule:The sum includes $r=0$. If the source is instead defined as the dimensionless quantity ${\beta }_{T}h$, the corresponding derivative omits the displayed inverse-temperature factor.

A real-space renormalization group step of length factor $b>1$ uses a local blocking kernel $B_b({\sigma }^{\prime },\sigma )\ge 0$ satisfying ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$. For example, a deterministic majority-spin rule assigns one coarse Ising spin to each block; tied blocks can be resolved with equal probabilities. Define the blocked statistical Hamiltonian byThe spacing becomes $a^{\prime }=ba$. All generated interactions, including multispin terms and the field-independent constant, must be retained for an exact transformation. Summing over ${\sigma }^{\prime }$ uses normalization to prove $Z(u^{\prime },C^{\prime },N^{\prime })=Z(u,C,N)$. Iteration gives $N_p=b^{-pD}N$ and $Z(u_p,C_p,N_p)=Z(u_0,C_0,N)$.

The coarse probability law is the exact marginal law of these block variables. Indeed, for any coarse observable $A$, its expected value is the microscopic expected value of ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )A({\sigma }^{\prime })$. A blocking rule that retains the long-wavelength order-parameter mode, with source and observable transformations tracked, therefore retains long-distance response and correlation functions. Their field normalization may change, while their physical length scales and critical singularities do not. Partition-function equality alone would not justify this assertion for an arbitrary kernel that discards all slow variables. Truncating the generated operator family is likewise an approximation.

For the free-energy recursion use dimensionless free energy per site,in the thermodynamic limit. Exact partition-function invariance gives $F(u_0,C_0)=b^{-pD}F(u_p,C_p)$. If one step sends $C$ to $C^{\prime }=b^{D}C+c(u)$, the coefficient $c(u)$ records the identity-operator contribution from eliminated fluctuations and normalization. ConsequentlyRepeated substitution derives the additive free-energy recursion under blocking:In physical energy units, divide this convention by ${\beta }_T$. The additive term is essential for absolute free energies even when it does not affect the leading critical powers.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. Linearize in scaling coordinates $v_i$ so that $v_i^{\prime }=b^{y_i}{v}_i+O(v^2)$. A relevant operator has $y_i>0$, an irrelevant operator has $y_i<0$, and a marginal operator has $y_i=0$ with behaviour decided by nonlinear terms. Thus the eigenvalues of the discrete linear map are $b^{y_i}$; the sign criterion applies to the generator exponents $y_i$, not to the discrete multipliers. The critical surface is the stable manifold on which every relevant scaling field has been tuned away. Irrelevant coordinates flow to the renormalization-group fixed point along it. A repulsive renormalization-group trajectory leaves the renormalization-group fixed point along an unstable relevant direction, ultimately entering a noncritical phase or another scaling regime.

To separate the singular free-energy density from its regular background, suppose the eliminated-mode contribution $g$ is analytic near the renormalization-group fixed point. Look for an analytic background $a(u)$ satisfying $a(u)=b^{-D}a(R_{b}u)+g(u)$. Then $F_s=f-a$ satisfies $F_s(u)=b^{-D}{F}_s(R_{b}u)$. In linear thermal and field coordinates, an analytic monomial $g_{mn}{t}^m{h}^n$ has background coefficientprovided the denominator is nonzero. This explicitly explains when analytic subtraction of an inhomogeneous renormalization recursion works. The printed inhomogeneous equation may describe the free-energy function before this subtraction, or a singular part defined only up to analytic terms. The inhomogeneous sum need not be small compared with the full free energy: an analytic constant can dominate a vanishing singular contribution. It can be omitted in deriving leading singular powers after regular subtraction, provided no renormalization-group free-energy resonance, marginal logarithm or dangerously irrelevant coupling obstructs the homogeneous critical asymptotics. At a renormalization-group free-energy resonance $m{\lambda }_t+n{\lambda }_h=D$, additive logarithms can remain, so simply dropping the term would miss them.

Suppose there are exactly two relevant scaling fields, thermal $t$ and magnetic $h$, with positive exponents ${\lambda }_t,{\lambda }_h$, and the remaining fields approach a regular limit. Physical reduced temperature and magnetic field agree with these coordinates to leading order, up to nonuniversal metric factors. The homogeneous scaling hypothesis for critical phenomena isas an asymptotic scaling equation. Choose $\ell =|t{|}^{-1/{\lambda }_t}$ to make the thermal argument $+1$ or $-1$. Defining $f_{\pm }(z)=F_s(\pm 1,z)$ givesThe two functions describe respectively $t>0$ and $t<0$: approaching the transition from disordered and ordered sides need not give the same function. In particular the ordered zero-field function has one-sided derivatives associated with spontaneous magnetization. Metric factors multiply this formula in different microscopic normalizations.

The correlation length in Bravais lattice units obeys $\xi (t,h)=\ell \xi ({\ell }^{{\lambda }_t}t,{\ell }^{{\lambda }_h}h)$. At $h=0$, the same choice of $\ell$ gives $\nu =1/{\lambda }_t$. Two thermal derivatives of $F_s(t,0)$ give $2-\alpha =D/{\lambda }_t$. One and two field derivatives give, respectively, $\beta =(D-{\lambda }_h)/{\lambda }_t$ and $\gamma =(2{\lambda }_h-D)/{\lambda }_t$, assuming the relevant branch amplitudes are nonzero. At $t=0$, choose $\ell =|h{|}^{-1/{\lambda }_h}$; then $M\propto \mathrm{sgn}(h)|h{|}^{(D-{\lambda }_h)/{\lambda }_h}$, so $\delta ={\lambda }_h/(D-{\lambda }_h)$. HenceSimple algebra now gives the Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ and the hyperscaling relation $\alpha =2-D\nu$. Also $\beta \delta ={\lambda }_h/{\lambda }_t=\beta +\gamma$, giving the Widom scaling relation. These statements refer to leading singular contributions with the homogeneous two-field scaling assumptions. Above the upper critical dimension, the quartic coupling is dangerously irrelevant because it is still needed to stabilize the ordered phase; ordinary mean-field critical exponents then satisfy Rushbrooke and Widom but generally not hyperscaling. Marginal logarithms also require qualifications at the upper critical dimension.

For the connected-correlation scaling form $G(r)=|r{|}^{-(D-2+\eta )}{f}_G(|r|/\xi )$, the function $f_G(x)$ tends to a finite nonzero value as $x\to 0$ and decays exponentially at large $x$, up to powers of $x$, in a massive short-range phase. The scaling ansatz alone does not fix that prefactor. If the usual massive Gaussian field correlation tail $G(r)\propto r^{-(D-1)/2}{e}^{-r/\xi }$ applies, matching the powers gives $f_G(x)\sim B{x}^{(D-3)/2+\eta }{e}^{-x}$, with a correlation-length-dependent amplitude in $G$ fixed by matching the critical scaling form.

In physical coordinates the lattice sum has site-density factor $a^{-D}$. Its long-distance singular part therefore giveswhere ${\Omega }_{D-1}$ is the area of the unit sphere in $D$ dimensions. For $\eta <2$, finite $f_G(0)$, an integrable exponential tail and a nonzero integral, the last factor tends to a finite nonzero constant as $\xi \to \infty$. Microscopic distances contribute a regular background. Thus ${\chi }_s\propto {\xi }^{2-\eta }\propto |t{|}^{-(2-\eta )\nu }$, proving the Fisher scaling relationEquivalently the field has scaling dimension $(D-2+\eta )/2$, so its conjugate field has ${\lambda }_h=(D+2-\eta )/2$, which gives the same identity from the exponent formulas above.

Let $o(\omega )$ be the number of open edges, $k(\omega )$ the number of connected components in the open subgraph, counting isolated vertices, and $m=|E|$. Summing the joint weight over spins requires a constant spin on every open component. There are exactly $q^{k(\omega )}$ choices. Thus the edge marginal isThis is the random-cluster measure, and the joint normalizing constant is $Z_{\mathrm{RC}}$.

Summing instead over edge configurations factors independently over edges. For a fixed spin configuration the resulting weight issince $1-p=e^{-\beta }$. The factor $e^{-\beta m}$ is independent of the spins. Hence the spin marginal is exactly the ferromagnetic Potts measure in the conventionIts partition function satisfies $Z_{\mathrm{RC}}=e^{-\beta m}{Z}_{\mathrm{P}}$.

Conditionally on $\sigma$, edge states are independent: an edge with different endpoint spins is closed with probability one, while one with equal endpoint spins is open with probability $p$. Conditionally on $\omega$, color its $k(\omega )$ open components independently and uniformly from $\{1,\dots ,q\}$. Equivalently, every compatible spin configuration has conditional probability $q^{-k(\omega )}$, and all incompatible configurations have probability zero. These are the two conditional laws of the Edwards-Sokal coupling.

If $x,y$ belong to the same open component, their colors agree surely; otherwise the two independent component colors agree with probability $1/q$. ThereforeRearranging proves the spin-connectivity identity for the Potts model:

Phase transitions refer to infinite-volume limits, not a single finite graph. The identity shows that connectivity is exactly the excess spin agreement above its independent-color baseline. More directly, use a wired random-cluster boundary condition and fix the corresponding boundary spin to color $1$. Its component has color $1$, while every finite component has a uniform color. In the compatible infinite-volume coupling,Consequently normalized spontaneous magnetization isThe onset of a wired infinite cluster and the onset of spontaneous magnetization in the Potts model are the same transition under $p=1-e^{-\beta }$, giving ${\beta }_c(q)=-\log (1-p_c(q))$. This also transfers any jump or continuity of that order parameter between the two descriptions; it makes no assumption about behavior exactly at the critical point.

The single-spin average in the effective field is $\tanh (\beta B_{\mathrm{eff}})$. Requiring it to equal the assumed average gives the mean-field self-consistency equationAt $B=0$, the slope of its right side at the origin is $2DJ/(k_{B}T)$. Thus $\boxed{T_c=2DJ/k_B}$. For $T>T_c$, $m=0$ is the only solution and is the stable minimum: thermal fluctuations dominate. For $T<T_c$, two stable nonzero solutions $\pm m(T)$ represent spontaneous magnetization, while the zero solution is unstable. At $T_c$ they merge continuously. Expanding $\tanh x=x-x^3/3+\cdots$ near the transition yields $m^2\sim 3(T_c-T)/T_c$, so the mean-field critical exponent is one-half. These are predictions of the approximation, not a proof of the exact Ising transition in every dimension.

Use inverse temperature ${\beta }_T=(k_{B}T{)}^{-1}$, reserving $\beta$ without a subscript for the critical exponent. Write $c=qJ$; each undirected bond occurs once, so the number of bonds is $Nq/2$. In the mean-field theory of the Ising model, write each Ising spin as ${\sigma }_i=m+\delta {\sigma }_i$ and neglect the product of fluctuations:The resulting independent-spin statistical Hamiltonian isThe constant corrects the double counting of interaction energy. The Ising spin sums now factorize and can all be evaluated:This is the approximate partition function at an assumed mean field; equilibrium fixes $m$ self-consistently. The corresponding Ising auxiliary mean-field free energy isIts equilibrium value gives the mean-field Helmholtz free energy in the imposed field. Away from a stationary point its parameter $m$ is an assumed field variable, not necessarily the actual mean Ising spin of that independent-spin distribution.

For the full small-$m$ expansion at fixed $h$, set $t_h=\tanh ({\beta }_{T}h)$ and $s_h={\mathrm{sech}}^2({\beta }_{T}h)$. Differentiating $\log \cosh z$ gives successive derivatives $t_h$, $s_h$, $-2t_h{s}_h$, and $2s_h(2-3s_h)$ at $z={\beta }_{T}h$. ThereforeAt zero field, spin inversion symmetry eliminates odd powers and this simplifies toThe quadratic coefficient changes sign and the quartic coefficient is positive atThus the zero-field mean-field prediction is a continuous, continuous phase transition: the stable zero spin magnetization develops two symmetry-related nonzero minima continuously below $T_c$.

Both requested routes give the same mean-field self-consistency equation. First, the one-spin expectation in the effective field isSecond, differentiating the Ising auxiliary mean-field free energy giveswhose stationary condition is precisely that mean-field self-consistency equation. Choose its stable, lowest-free-energy branch rather than every algebraic solution.

An equally useful mean-field approximation parametrizes the trial distribution by its actual mean Ising spin. Its probabilities are $(1\pm m)/2$, giving the Bragg-Williams free energy of the Ising modelThis energy-minus-entropy function has expansionIts stationary equation is $h=k_{B}T\mathrm{arctanh}m-cm$, again equivalent to the mean-field self-consistency equation. The two functions differ away from equilibrium, but agree on stationary branches: for $z={\beta }_T(h+cm)$ and $m=\tanh z$, the entropy bracket equals $mz-\log (2\cosh z)$. Their small-$m$ coefficients therefore need not agree at arbitrary temperature; at $T_c$ they have the same leading critical quartic coefficient $c/12$. This distinction prevents confusing the auxiliary-field expansion with the physical-magnetization variational expansion.

For the order-parameter critical exponent, expand the equation of state at $h=0$:On a nonzero stable branch,Hence $\beta =1/2$. The spontaneous magnetization is understood by selecting a branch with an infinitesimal field after the thermodynamic limit; a finite symmetric sample has zero exact zero-field mean Ising spin.

For the magnetic susceptibility per site, define $\chi =\partial m/\partial h$ with $h$ in energy units. Implicit differentiation givesAbove $T_c$, $m=0$ and $\chi =1/[k_B(T-T_c)]$. Below $T_c$, evaluated on a selected ordered branch, the expansion of $m^2$ gives $\chi \sim 1/[2k_B(T_c-T)]$. Thus the magnetic-susceptibility critical exponent is $\gamma =1$ on both sides, with different amplitudes. At $T=T_c$, the equation of state becomesso the critical-isotherm exponent is $\delta =3$. These are mean-field critical exponents, not a claim that neglecting fluctuations gives the exact Ising transition in every dimension.

In the Landau theory approximation, choose the global minimum of the Landau free energy as a function of the magnetization. Since ${\mathcal{A}}_6>0$, the polynomial is a coercive function and a global minimum exists. The candidates are stationary points satisfyingCheck their values of the free-energy density; a stationary point or a metastable local minimum alone need not be the equilibrium state. Positive curvature ${\mathcal{A}}_2+3{\mathcal{A}}_4{m}^2+5{\mathcal{A}}_6{m}^4$ ensures local stability; when it vanishes, higher terms must be checked. The selected global minimum is the equilibrium magnetization within this approximation.

At zero magnetic field, equal opposite minima describe distinct phases exhibiting spontaneous symmetry breaking. A spontaneous magnetization is obtained by taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$. A symmetric finite-volume average can instead have zero expectation even when the minima are nonzero. Thus one must specify the phase when interpreting the minimizer as the equilibrium expectation.

The spin-length sum rule and Fourier inversion giveThe zero mode contains the square of the field-directed magnetization, while the infrared bound controls all nonzero modes. After $L\to \infty$ and then $h\downarrow 0$,Near zero, $\epsilon (k)\asymp |k{|}^2$, so the integral is finite exactly when $d\ge 3$. Choose ${\beta }_0$ larger than the resulting finite constant. Then for $\beta >{\beta }_0$ the right side is positive, and $m(\beta )\ge c>0$.

If $h\downarrow 0$ before $L\to \infty$, every finite torus retains global $O(n)$ symmetry and its magnetization is zero. The reversed iterated limit is therefore zero. The order of limits is what permits spontaneous magnetization.

Replace the applied field in the doublet response by the Weiss molecular field $B_E={\mu }_0\lambda M$. At zero applied field,The equation is odd in $M$, so the two ordered orientations have equal magnitude. With $m=|M|/(n{\mu }_B)$, defineThe mean-field self-consistency equation becomesPlot $y=m$ and $y=\tanh (m/t)$. The second has slope $1/t$ at zero and is concave for positive $m$. For $0<t<1$ it starts above the straight line but is below it by $m=1$, giving one positive intersection. For $t\ge 1$, $\tanh (m/t)<m$ for every $m>0$. HenceAt $T=0$ the limiting solution is $m=1$. The onset of spontaneous magnetization is the mean-field Curie temperature.

The small-argument Taylor series of the single-site response isLet $K=4\beta J$. For a small nonzero order parameter at zero field, the self-consistency equation becomesSince $K_c=3/2$ and $K=K_c/(1+t)$, where $t=T/T_c-1$, this givesConsequentlyThe positive branch is the expression printed in the question, while its negative partner is required by spin inversion symmetry. The square-root behavior gives the mean-field critical exponent $1/2$ for the spontaneous magnetization.

For attractive nearest-neighbour interactions, $J>0$, the transformed model is a ferromagnetic Ising model. In three dimensions and at $T<T_c$, its infinite-volume spontaneous magnetization has two branches. As the effective field crosses zero, the equilibrium mean spin changes discontinuously from a negative value to a positive value. Since $(\partial h/\partial \mu {)}_T=1/2$, varying the physical chemical potential crosses this same first-order phase transition.

The particle-hole-symmetric coexistence line isThe mean Ising spin is a rescaled occupation density. Writing $n=⟨N⟩/V$ for the physical number density,The negative-spin phase is a dilute gas and the positive-spin phase is a dense, liquid-like phase. If the spontaneous spin magnitude at coexistence is $m_0(T)>0$, their number densities areThus the physical order parameter is a density or occupation imbalance, not an actual magnetic moment of these particles. In a closed system with an intermediate mean density, the same transition appears as phase coexistence between the two densities. At $T=T_c$ the density contrast vanishes continuously; above $T_c$, sweeping the chemical potential gives no coexistence jump. A true discontinuity requires the thermodynamic limit; a finite sum over finitely many cells remains analytic.

Given a random-cluster configuration, its components receive independent uniform colors under the Edwards-Sokal coupling. Two vertices have equal Potts spins with probability $1$ if connected and $1/q$ otherwise. Averaging proves the identity. With a wired component fixed to color $1$, the same argument gives normalized spontaneous magnetization $(q{\pi }^1({\sigma }_0=1)-1)/(q-1)={\varphi }^{\mathrm{w}}(0\leftrightarrow \infty )$ in infinite volume. Thus its onset is the wired random-cluster critical probability, transformed to ${\beta }_c=-\log (1-p_c)$ in this convention.

At zero field and with spin inversion symmetry in its boundary conditions, a finite Ising model pairs every configuration with its spin-reversed configuration. Their equal Boltzmann weights force the expected value of its spin magnetization $m=N^{-1}{\sum }_i{s}_i$ to vanish. Below a model's ordering critical temperature, an arbitrarily small positive field can instead select the positive spontaneous magnetization if size is taken to infinity first:The distinction is the order of limits and the stated symmetric boundary conditions; plus boundary conditions already break spin inversion symmetry. A zero-field symmetric mixture of the two ordered states can retain zero mean spin magnetization even in the thermodynamic limit. Also, $⟨|m|⟩>0$ in a finite system is not by itself proof of spontaneous symmetry breaking.^[ref]