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In a smooth scalar Landau free energy, a line of coexistence between two distinct minima can end by those minima and the intervening maximum coalescing. At a generic endpoint with no competing third phase,The first condition is stationarity, the second removes the inverse susceptibility, and the third is needed because nonzero cubic curvature at a zero-quadratic point would prevent it being a stable minimum. After shifting $M=M_c+\psi$ and mixing two control parameters, the local potential is $V_c+R{\psi }^2/2+U{\psi }^4/4-H\psi +\cdots$, $U>0$. Its two-phase coexistence line $H=0,R<0$ reaches $R=H=0$, with $\Delta M=2\sqrt{-R/U}\to 0$ and diverging magnetic susceptibility. This is the familiar critical endpoint of a quartic Landau free energy.

The word “must” needs the two-phase endpoint assumptions. A first-order line can instead end at a triple point or a boundary of the available control parameters. There is an explicit scalar counterexample in the sextic potential: fix $u<0<v$ and vary $(r,h)$. The $+M/-M$ coexistence line at $h=0$, $r<3u^2/(16v)$ ends when the $M=0$ phase also becomes globally stable. At that triple point the nonzero minima remain $\pm \sqrt{-3u/(4v)}$; their jump does not vanish. Thus the literal unrestricted assertion is false. The continuous-endpoint argument above establishes the intended case and distinguishes it from this counterexample.

Along an isolated two-phase coexistence line, two distinct stable minima have equal free energy. If this line ends at an ordinary interior endpoint without encountering another phase, the distinction between the two phases must disappear: their minima and the intervening barrier merge. For separated nondegenerate minima, their stationary values are smooth functions of the controls. The difference $\Delta V=V(m_1)-V(m_2)$ has ${\partial }_h\Delta V=-m_1+m_2\ne 0$. The implicit function theorem therefore continues the equation $\Delta V=0$ as a local coexistence curve. It cannot end while those distinct stable minima persist.

At the merger the potential satisfiesThe third-derivative condition is essential: a stationary zero-curvature point with nonzero third derivative is an inflection, not a stable equilibrium. Shifting to $\psi =m-m_c$ gives an ordinary quartic Landau free energy with two independent control directions, temperature-like and field-like. The curvature and the order-parameter discontinuity vanish, the magnetic susceptibility diverges, and the correlation length diverges. The endpoint is therefore associated with a continuous transition. This is the critical endpoint of a quartic Landau free energy mechanism.

In the symmetric quartic example, the coexistence line is $h=0,r<0$. Its jump $2\sqrt{-r/u}$ vanishes at $(r,h)=(0,0)$, where $V=u{m}^4/4$. This supplies a concrete realization. The word “must” requires the stated ordinary-endpoint assumptions: a first-order line can instead meet a third phase, a multicritical point, a boundary of parameter space, or continue without a finite endpoint. A spinodal point alone is not an equilibrium critical endpoint.

The scaling hypothesis for critical phenomena says that the singular part of the equilibrium free energy is a generalized homogeneous function of its thermal and field-like controls. Smooth backgrounds must first be removed; the hypothesis does not claim that the entire free energy, including arbitrary regular terms, has a pure scaling form.

For the ordinary scalar quartic LG theory, set $r=r_{t}t$ with $r_t>0$ and $u>0$, and rescale the uniform order parameter asThe order-parameter-dependent free-energy density becomesDefine $f_{\pm }(H)$ as the minimum of the braces, with the sign of the quadratic term respectively positive or negative. ConsequentlyFor an extensive $A_s$, $a$ additionally contains the system volume. The printed double-inequality subscript labels the two temperature branches: the upper-temperature function is $f_{+}$ and the lower-temperature function is $f_{-}$. It does not classify positive and negative magnetic fields. In particular $f_{+}(0)=0$ and $f_{-}(0)=-1/4$; below the transition the field dependence has a cusp at zero, so derivatives are taken on a selected branch.

For the following derivatives, take $A_s$ to be a density, so $m$ is the magnetization density; for total free energy the derivative gives total magnetization instead. Differentiate this mean-field scalar free-energy scaling form. The magnetization is $m=-\partial A_s/\partial h=-ab|t{|}^{1/2}{f}_{\pm }^{\prime }(H)$, giving $m_0\propto (-t{)}^{1/2}$ and $\beta =1/2$. The magnetic susceptibility is $\chi =-{\partial }_h^2{A}_s=-a{b}^2|t{|}^{-1}{f}_{\pm }^{\prime \prime }(H)$, giving $\gamma =1$. At zero field the nonzero curvature amplitudes are $-f_{+}^{\prime \prime }(0)=1$ and $-f_{-}^{\prime \prime }(0^{+})=1/2$.

To allow nonclassical critical exponents, replace the fixed powers byThe heat-capacity critical exponent is defined by $C_{V,s}\sim |t{|}^{-\alpha }$; temperature differentiation gives the thermal exponent $2-\alpha$ in $A_s$. The order-parameter critical exponent has $m_0\sim (-t{)}^{\beta }$, and the magnetic-susceptibility critical exponent has $\chi \sim |t{|}^{-\gamma }$. Differentiating the scaling form givesEliminating $\Delta$ proves the Rushbrooke scaling relationThe critical-isotherm exponent is defined by $m(0,h)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. At fixed small $h$, the $t\to 0$ limit requires $f_{\pm }(H)\sim c|H{|}^{(2-\alpha )/\Delta }$, so that the temperature factors cancel. Therefore $1/\delta =(2-\alpha -\Delta )/\Delta =\beta /\Delta$, yielding $\Delta =\beta \delta$. But the two differentiated identities also give $\Delta =\beta +\gamma$, and hence the Widom scaling relationThese are relations among the leading power indices. At marginal dimensions, multiplicative logarithms can accompany them, and an additive analytic background must not be mistaken for the singular scaling contribution.

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.

These are mean-field critical exponents at a tricritical point. Hold the other control at its tricritical value and take the coefficients to be real analytic functions of temperature along a thermal approach with ${\mathcal{A}}_2=a(T-T_c)+o(T-T_c)$, $a>0$, ${\mathcal{A}}_6\to s>0$, and ${\mathcal{A}}_4=0$ or ${\mathcal{A}}_4=o(|{\mathcal{A}}_2{|}^{1/2})$. At fixed tricritical $g$, a generic real analytic temperature path with nonzero thermal slope has ${\mathcal{A}}_4=O(T-T_c)$ and meets the last condition. Paths tangent to a scaling direction can have different apparent temperature powers.

At zero field on the ordered side with ${\mathcal{A}}_4=0$, stationarity gives $m^4=-{\mathcal{A}}_2/s$; under the more general approach ${\mathcal{A}}_4=o(|{\mathcal{A}}_2{|}^{1/2})$, this relation holds to leading order. Thus the order-parameter critical exponent isFor the zero-quartic representative, putting this minimum into the Landau free energy gives the singular order-parameter contribution (and the leading asymptotic contribution under the stated negligible-quartic approach)Using $C_{\mathrm{s}}=-T{\partial }_T^2{f}_{\mathrm{s}}$ at fixed other control yields $C_{\mathrm{s}}\sim (T_c-T{)}^{-1/2}$, specifically $T{a}^{3/2}/(4\sqrt{s(T_c-T)})$ to leading order. HenceThis heat-capacity critical exponent refers to the nonzero ordered-side singular amplitude; in this mean-field approximation to Landau theory, the disordered global minimum contributes zero singular free energy, while any regular background remains. Fluctuation corrections lie outside the present approximation.

In the disordered phase, differentiating the equation of state at $m=h=0$ gives the magnetic susceptibility $\chi =1/{\mathcal{A}}_2$. On the ordered branch its curvature is ${\mathcal{A}}_2+5s{m}^4=4|{\mathcal{A}}_2|$, so $\chi =1/(4|{\mathcal{A}}_2|)$. Both give the magnetic-susceptibility critical exponentAt the tricritical temperature with both lower coefficients zero, $h=s{m}^5$, so $m=\mathrm{sgn}(h)(|h|/s{)}^{1/5}$ and the critical-isotherm exponent isThe Widom scaling relation $\gamma =\beta (\delta -1)$ and Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ both check these powers. Their validity here is a mean-field conclusion, not a claim that fluctuations in every spatial dimension preserve them.

The microcanonical ensemble assigns equal probability to accessible quantum states in a narrow energy shell of an isolated system with fixed energy, particle number and external parameters. To derive the canonical ensemble, put a small system in weak thermal contact with a much larger isolated bath. If the small-system state has energy $E_i$, its probability is proportional to the number ${\Omega }_b(E_{\mathrm{total}}-E_i)$ of bath states. Expanding the bath entropy gives $S_b(E_{\mathrm{total}}-E_i)=S_b(E_{\mathrm{total}})-E_i/T+\cdots$, since $\partial S_b/\partial E=1/T$. Negligible interaction energy and an effectively constant bath temperature therefore give $p_i=e^{-E_i/(k_{B}T)}/Z$. The microcanonical ensemble suits an isolated fixed-energy system; the canonical ensemble suits a system exchanging energy with a thermostat at fixed temperature, with particle number still fixed.

For one atom, summing the finite geometric series over $m=-J,\dots ,J$ gives the partition functionAt $x=0$ the ratio is understood by continuity as $2J+1$. Logarithmic differentiation gives the mean magnetic dipole momentThis is odd in $x$, has small-$x$ expansion ${\mu }_{B}J(J+1)x/3+O(x^3)$, and saturates at $\pm J{\mu }_B$. The requested common graph is plotted in units of ${\mu }_B$: its ordinate is $⟨{\mu }_z⟩/(J{\mu }_B)$, so multiplication by ${\mu }_B$ gives $⟨{\mu }_z⟩/J$. The normalized slopes at zero are $(J+1)/3$ and the limits are $\pm 1$.

The magnetic susceptibility isEquivalently it is $N{\mu }_B^2\mathrm{Var}(m)/(k_{B}T)\ge 0$. For $k_{B}T\gg {\mu }_B|B|$, the apparent singular terms cancel, yielding Curie lawFor fixed nonzero $B$, $T\to 0$ and integer $J\ge 1$, the first excited magnetic level is separated from the aligned ground state by ${\mu }_B|B|$. Its exponentially small occupation givesThe moments are saturated, so changing the magnetic field hardly changes their alignment. This is a fixed-nonzero-field low-temperature limit. At exactly $B=0$, differentiating first gives the Curie expression for every $T>0$, which diverges as $T\to 0$; the two limits do not commute in this noninteracting model. For $J=0$ both moment and susceptibility vanish identically.

At the tricritical point, $a_4=0$ and the zero-field Landau free energy is $f=a_2{m}^2+a_6{m}^6$. Below $T_c$, minimization givesSince $a_2\sim T-T_c$, the order-parameter critical exponent isAt this minimum, $f_{\mathrm{sing}}=a_2{m}^2+a_6{m}^6$ is proportional to $-|T-T_c{|}^{3/2}$. Comparing this with $f_{\mathrm{sing}}\sim |T-T_c{|}^{2-\alpha }$ gives the heat-capacity critical exponentAfter adding the magnetic contribution $-Bm$, the inverse zero-field magnetic susceptibility is the curvature $f^{\prime \prime }(m)$. Above $T_c$ it is $2a_2$, while below $T_c$ it is $-8a_2$, so $\chi \sim |T-T_c{|}^{-1}$ and the magnetic-susceptibility critical exponent isFinally, exactly at $T_c$ the equation of state is $B=6a_6{m}^5$. Hence $m\sim B^{1/5}$ and the critical-isotherm exponent is

Let $g(E)$ denote the total density of states, including both spin directions. The field lowers one spin energy by ${\mu }_{B}B$ and raises the other by the same amount. To first order in $B$, each spin population changes by the one-spin density $g(E_F)/2$ times the displacement of its Fermi edge. The population imbalance is thereforeMultiplying by the Bohr magneton carried by each excess aligned electron givesThus, with this total-density convention,The zero-temperature magnetic susceptibility isThis weak positive response is Pauli paramagnetism. If susceptibility per unit volume is required, both $M$ and $g$ are divided by $V$.

Put $t=T-T_c$, so $a_4\sim t$. At zero field the stationary equation factors asFor $t>0$, $m=0$ is the unique minimum. For $t<0$, it is unstable and the two minima areThe order parameter therefore tends continuously to zero, and its order-parameter critical exponent is $\boxed{\beta =1/2}$.

At either ordered minimum the singular free-energy density iswhereas it is zero for $t>0$. Two temperature derivatives give a singular heat capacity proportional to $|t|$ below the transition and zero above it, so the heat-capacity critical exponent is $\boxed{\alpha =-1}$.

The inverse magnetic susceptibility at a stable minimum is the curvature ${\chi }^{-1}={\partial }^{2}f/\partial m^2$. Below $T_c$,and hence ${\chi }_{-}\sim |t{|}^{-2}$ and $\boxed{{\gamma }_{-}=2}$. Above $T_c$, however, the curvature at $m=0$ vanishes for every $t>0$. Indeed, at small field $B=4a_4{m}^3+O(m^5)$, so $m\sim (B/(4a_4){)}^{1/3}$ and the linear susceptibility is already infinite away from the critical point. Consequently the usual magnetic-susceptibility critical exponent ${\gamma }_{+}$ is not defined for this exceptional free energy; assigning it a finite value would incorrectly assume a quadratic term.

At $T=T_c$, the equation of state is $B=6a_6{m}^5$, so $m\sim |B{|}^{1/5}$ and the critical-isotherm exponent is $\boxed{\delta =5}$. Thus the transition is continuous, although its missing quadratic term makes the high-temperature linear response singular throughout that phase.

Write $t=T-T_c$, take ${\alpha }_2=at$ with $a>0$, and include the magnetic term $-Bm$. At $B=0$, minimizing the Landau free energy gives $m=0$ for $t>0$ andfor $t<0$, so the order-parameter critical exponent is $\beta =1/2$. Substitution gives a singular free-energy density proportional to $-t^2$ below $T_c$, whose second temperature derivative has a finite jump, so the heat-capacity exponent is $\alpha =0$. Above $T_c$, the equation ${\alpha }_{2}m+{\alpha }_4{m}^3=B$ gives the magnetic susceptibility $\chi =(\partial m/\partial B{)}_{B=0}=1/{\alpha }_2\sim t^{-1}$, hence $\gamma =1$. At $t=0$, $B={\alpha }_4{m}^3$, so $m\sim B^{1/3}$ and $\delta =3$.

A smooth scalar Landau free energy can end a two-phase phase coexistence curve by merging two minima and their intervening maximum. At a generic stable endpoint, its first three derivatives vanish and its fourth derivative is positive. A field shift and two mixed control parameters reduce the local form to a quartic Landau free energy; the order-parameter jump then vanishes and the magnetic susceptibility diverges. This conclusion assumes there is no competing third phase and that the endpoint lies inside the control-parameter domain. It is not a general theorem about every first-order line: a sextic potential with fixed negative quartic coefficient has a symmetry-related ordered coexistence line that ends at a triple point with a finite jump.