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With $J=4$, the measured effective paramagnetic moment gives the Landé g-factorFor the free-ion Hund's rules values $S=1$, $L=5$, $J=4$, the stated formula yieldsThusThe source fit is consistent with the unquenched $4f$ orbital magnetic moment of the free praseodymium ion, within graphical accuracy. Small deviations need not indicate a change of electron configuration: crystal field splitting, interactions and the fit range can affect a measured magnetic susceptibility.

The $1/x$ terms in the Brillouin function cancel. Keeping the linear terms givesConsequently,Use the dimensionless SI magnetic susceptibility $\chi =\partial M/\partial H_{\mathrm{applied}}$ and $B_{\mathrm{applied}}={\mu }_0{H}_{\mathrm{applied}}$, as in the source's molecular-field convention. The Curie law isIf susceptibility is instead defined as $\partial M/\partial B$, its coefficient is $C/{\mu }_0$ and is dimensionful. The SI convention above makes the subsequent Curie–Weiss law and $\lambda$ consistent.

A further derivative gives the magnetic susceptibility within a selected pure thermodynamic phase:Assuming a finite nonzero zero-field amplitude on the branch considered, comparison with $\chi (t,0)\asymp |t{|}^{-\gamma }$ givesTogether with part (a), this proves the Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ and the Widom scaling relation $\gamma =\beta (\delta -1)$. Below the transition, use the one-sided response of an ordered phase; differentiating through the cusp of the equilibrium free energy at $h=0$ would include phase switching rather than this critical response.

First interpret the displayed critical correlation function as the connected correlation functionAbove the transition at zero field the subtraction is zero; in an ordered phase it is essential. For a dimensionless source coupled to the total magnetization $M=\int d^{d}xm(x)$, differentiating $Z$ twice gives $\chi =V^{-1}(⟨M^2⟩-⟨M{⟩}^2)$. Translation invariance then proves the correlation-function susceptibility sum ruleInsert the scaling form, use spherical coordinates and set $s=r/\xi$:For $\eta <2$, finite nonzero $g(0)$ and an integrable large-$s$ tail, the last integral tends to a finite nonzero constant. Microscopic separations add an analytic response background. Thus the singular magnetic susceptibility obeys ${\chi }_s\asymp {\xi }^{2-\eta }$, and at zero field $\xi \asymp |t{|}^{-\nu }$ gives the Fisher scaling relationA dimensional magnetic field would add a nonsingular inverse-temperature prefactor. Integrating the full unsubtracted ordered-phase correlation function instead would yield a volume-divergent term $V⟨m{⟩}^2$, which is not the intrinsic susceptibility appearing in this exponent identity.

Critical exponents describe leading singular powers near a continuous phase transition. Using the reduced temperature $t=(T-T_c)/T_c$, the standard definitions includeThese are the order-parameter critical exponent, magnetic-susceptibility critical exponent, heat-capacity critical exponent and correlation-length critical exponent. The critical-isotherm exponent is defined by $M(0,H)\sim \mathrm{sgn}(H)|H{|}^{1/\delta }$. The anomalous dimension $\eta$ gives the critical long-distance connected correlation function, $G_c(x)\sim |x{|}^{-(d-2+\eta )}$, when an ordinary scale-invariant critical point exists.

In the Landau approximation, minimize the Landau free energy to obtain the equation of state, differentiate it with respect to the field conjugate to an order parameter to obtain magnetic susceptibility, and differentiate the minimized free energy with respect to temperature to obtain heat capacity. These operations determine $\beta ,\delta ,\gamma ,\alpha$, respectively. Expanding the Landau-Ginzburg theory about its uniform minimum gives the quadratic fluctuation kernel $K{q}^2+V^{\prime \prime }(M)$, whose Fourier transform determines the correlation length and the anomalous dimension. At the ordinary quartic critical point this gives ${\xi }^2=K/V^{\prime \prime }(M)\propto |t{|}^{-1}$ and $\eta =0$.

For $u>0$, $r=0$ gives $H=u{M}^3$ and hence $\delta =3$; the equilibrium free energy $-r^2/(4u)$ gives a finite heat capacity discontinuity, conventionally $\alpha =0$. Together with the explicit calculation below, the ordinary mean-field critical exponents areThese describe neglect of fluctuation corrections, rather than the exact exponents of every low-dimensional Landau-Ginzburg theory.

A tricritical point is the meeting of a line of continuous phase transitions and a line of first-order phase transitions. In the symmetric scalar Landau free energy, it occurs at $r=u=0$ with $v>0$. Two independent controls are needed to tune both coefficients. The continuous transition lies at $r=0,u>0$, and the tricritical three-phase line found above has $r=3u^2/(16v),u<0$; both approach the tricritical point. At zero conjugate field the three coexisting states on this first-order line are the disordered state and the two opposite ordered states.

Setting $u=H=0$ makes the leading stabilizing interaction sextic. The nonzero minimum obeys $M^4=-r/v$, so the order-parameter critical exponent along this tuned direction is ${\beta }_{\mathrm{tri}}=1/4$. Substitution givesand two temperature derivatives give the heat-capacity critical exponent ${\alpha }_{\mathrm{tri}}=1/2$. At $r=u=0$, the equation of state is $H=v{M}^5$, giving the critical-isotherm exponent ${\delta }_{\mathrm{tri}}=5$.

For magnetic susceptibility, differentiating $H=rM+v{M}^5$ gives ${\chi }^{-1}=r+5v{M}^4$. This is $r$ above the transition and $-4r$ below, hence ${\gamma }_{\mathrm{tri}}=1$. The quadratic fluctuation kernel still has a mass proportional to $|r|$, giving ${\nu }_{\mathrm{tri}}=1/2$ and ${\eta }_{\mathrm{tri}}=0$. Thus the mean-field tricritical exponent calculation yieldsDetuning $u$ gives tricritical crossover scaling; one must specify the approach to the tricritical point when quoting these critical exponents. The sextic interaction has upper critical dimension three, so sufficiently strong long-distance fluctuations change these mean-field predictions below three dimensions.

Take the symmetric scalar Landau free energy with $u>0$, $r=at$ for a positive constant $a$, and leading equation of stateAt zero conjugate field, minimization gives $M=0$ for $t>0$ and $M=\pm \sqrt{-r/u}$ for $t<0$. Selecting the positive pure thermodynamic phase,Consequently the order-parameter critical exponent is $\boxed{\beta =1/2}$.

Differentiate the equation of state at fixed temperature, within the selected phase:Both sides diverge as $|t{|}^{-1}$, so the magnetic-susceptibility critical exponent is $\boxed{\gamma =1}$. The leading above-to-below magnetic susceptibility amplitude ratio is two. These are the ordinary mean-field critical exponents, valid under the fluctuation qualification explained above.

The correlation length is the distance over which fluctuations remain appreciably correlated. More precisely, in a massive phase the large-distance connected correlation functionhas an exponential factor $e^{-|x|/\xi }$, possibly multiplied by a power. Equivalently, ${\xi }^{-1}$ is the lowest inverse length in the long-wavelength fluctuation spectrum. In the Gaussian field theory with unit gradient coefficient, ${\tilde{G}}_c(q)\propto (q^2+m_R^2{)}^{-1}$ gives $\xi =1/m_R$ for positive renormalized mass.

At an ordinary continuous phase transition, the restoring force for long-wavelength order-parameter fluctuations vanishes and there is no finite correlation scale. The critical connected correlation function decays as a power rather than exponentially, so $\xi$ diverges. The correlation-function susceptibility sum rule relates magnetic susceptibility to the integral of this connected correlation function; its divergence is consistent with correlations extending over increasingly large distances. Interactions change the powers, so the renormalized mass and the bare mass need not vanish at the same temperature.

Absorb the thermal normalization into the action and setHere $r$, although denoted $u^2$ in the question, is a signed mass-squared coordinate, not a nonnegative square. This matters because its interacting fixed-point value is negative. Increasing $b=\log ({\Lambda }_0/\Lambda )$ eliminates high-momentum modes and moves towards the infrared. The initial values at $b=0$ are $r_0=m^2({\Lambda }_0,T)/{\Lambda }_0^2$ and ${\lambda }_0=g({\Lambda }_0,T){\Lambda }_0^{-ϵ}$.

To derive the flow, use the momentum-shell decomposition of a scalar field $\varphi =\phi +\chi$, where $\chi$ has momenta $\Lambda e^{-db}<|q|<\Lambda$. Its Gaussian shell covariance is $(q^2+m^2{)}^{-1}$. Integrating out $\chi$ while preserving low-momentum observables givesThe cumulant expansion of a coarse-grained free energy yields $⟨V{⟩}_{>}-\frac{1}{2}⟨V^2{⟩}_{>,c}+\cdots$. By Wick theorem,The last term is field independent, while the quadratic term gives the one-loop shell mass renormalization in scalar quartic theoryThis tadpole is independent of the external momentum, so no gradient renormalization occurs at this order.

For the quartic correction, the mixed vertex is $(g/4)\int {\phi }^2{\chi }^2$. Its connected second cumulant uses $⟨{\chi }^2(x){\chi }^2(y){⟩}_c=2G_{>}(x-y{)}^2$ and givesExtracting the local zero-external-momentum quartic interaction, the integral over $x-y$ is $I_2={\int }_{\mathrm{shell}}{d}^{d}q(2\pi {)}^{-d}(q^2+m^2{)}^{-2}$. Comparing its coefficient with $g{\phi }^4/4!$ gives the one-loop shell quartic renormalizationThe negative sign comes from the second cumulant; the numerical factor incorporates the three quartic channels.

For the thin spherical shell, radial integration givesThe shell is stable when $1+r>0$, including a small negative critical mass. Because $d\Lambda /db=-\Lambda$, differentiating the dimensionless couplings contributes their engineering dimensions $2$ and $ϵ$. Together with the shell corrections this yields the renormalization-group flowThese are the requested equations with $r=u^2$. They express independence of the long-distance theory from the arbitrarily chosen ultraviolet cutoff, to the retained perturbative order; higher interactions and gradient corrections enter at higher orders.

Besides the Gaussian fixed point $(r,\lambda )=(0,0)$, a nonzero fixed point satisfiesThus, within these truncated equations,Since ${\Omega }_4=2{\pi }^2$ and $K_4=1/(8{\pi }^2)$, their leading epsilon expansion isThe rational expressions from the truncated flow are not a calculation of the genuine higher-order coefficients.

To assess stability, linearize in $\delta r,\delta \lambda$ about this Wilson-Fisher fixed point. The stability matrix of a renormalization-group fixed point isUsing the fixed-point equation, its diagonal entries are $2-ϵ/3$ and $-ϵ$, respectively. Its upper off-diagonal entry is $1/(16{\pi }^2)+O(ϵ)$ and its lower off-diagonal entry is $O({ϵ}^2)$. Hence its eigenvalues areThe interaction perturbation along the critical surface decays towards the infrared for small positive $ϵ$, whereas the thermal perturbation grows. Therefore infrared attraction requires tuning to the critical surface. The source's unqualified wording cannot mean attraction in both coupling directions: the positive thermal eigenvalue is precisely what produces the divergence of the correlation length. This is the thermal relevant direction at the Wilson-Fisher fixed point.

Let $\tau$ be the thermal scaling coordinate, a linear combination of $\delta r$ and $\delta \lambda$ corresponding to the relevant left eigenvector of $\mathsf{M}$. To leading order, $\tau =\delta r+[1/(32{\pi }^2)+O(ϵ)]\delta \lambda$ is possible. Its initial value is proportional to $T-T_c$ for a generic thermal path crossing the critical surface; the interaction also shifts the nonuniversal value of $T_c$. In the linear regime,Choose $b_{\ast }$ so that $|\tau (b_{\ast })|$ is of order one. At this scale the dimensionless correlation length is finite. Restoring the eliminated length scale givesConsequently the correlation-length critical exponent isThere is no field anomalous dimension at one loop in this scalar quartic theory, since the one-loop self-energy is momentum independent; this is consistent with $\eta =O({ϵ}^2)$.

The scaling hypothesis for critical phenomena describes the singular part of the equilibrium free-energy density, rather than its analytic background. For an isotropic system in spatial dimension $D$, with thermal scaling field $t$ and conjugate field $h$, writeThis follows from coarse graining near a renormalization-group fixed point: a correlation volume is enlarged by $b^D$, while the relevant thermal and field perturbations grow with their eigenvalues. Choosing $b=|t{|}^{-1/y_t}$ givesThe two functions describe the two sides of the transition. Nonuniversal metric factors multiplying $t,h,f_s$ must be allowed when comparing microscopic models. Exponents and suitably normalized scaling functions or amplitude ratios characterize a universality class.

Differentiating the free-energy density gives the order parameter $M=-{\partial }_h{f}_s$, magnetic susceptibility $\chi ={\partial }_{h}M$ and singular heat capacity proportional to $-{\partial }_t^2{f}_s$. ThusThe subscript distinguishes the order-parameter exponent from inverse temperature. At $t=0$, choose $b=|h{|}^{-1/y_h}$ to obtain $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, with $\delta =y_h/(D-y_h)$. These derivatives imply the Rushbrooke scaling relation $\alpha +2{\beta }_{\mathrm{mag}}+\gamma =2$ and the Widom scaling relation $\gamma ={\beta }_{\mathrm{mag}}(\delta -1)$. Equivalently, the equation of state has $h=|t{|}^{\Delta }{\mathcal{H}}_{\pm }(M/|t{|}^{{\beta }_{\mathrm{mag}}})$, with $\Delta ={\beta }_{\mathrm{mag}}\delta ={\beta }_{\mathrm{mag}}+\gamma$.

The correlation length scales as $\xi \sim |t{|}^{-\nu }$, $\nu =1/y_t$. At criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$, giving $y_h=(D+2-\eta )/2$. Integrating $G$ over a correlation volume gives $\chi \sim {\xi }^{2-\eta }$ and hence the Fisher scaling relation $\gamma =(2-\eta )\nu$. The hyperscaling relation $2-\alpha =D\nu$ expresses one singular free-energy contribution per correlation volume.

These laws organize measurements: plotting $M/|t{|}^{{\beta }_{\mathrm{mag}}}$ against $h/|t{|}^{\Delta }$ collapses near-critical curves onto the appropriate scaling function. A finite system replaces the divergent length by its size $L$; the corresponding prediction is $f_s=L^{-D}\mathcal{F}(t{L}^{y_t},h{L}^{y_h})$ under the same scaling assumptions. Finite size rounds a sharp thermodynamic singularity.

The hypothesis has a domain of validity. At a renormalization-group fixed point one generally needs $f_s=b^{-D}{f}_s(b^{y_t}t,b^{y_h}h,b^{y_u}u,\dots )$, including other scaling fields. Ordinary irrelevant couplings can be set to zero in a nonsingular limiting scaling function. A dangerously irrelevant coupling cannot: above the upper critical dimension of scalar quartic theory, the quartic term is irrelevant at the Gaussian point but still stabilizes the ordered phase. Consequently naive hyperscaling fails there even though other exponent relations can remain valid. Marginal variables can produce logarithms, anisotropic systems require the appropriate volume exponent, and the Berezinskii–Kosterlitz–Thouless transition has an essential correlation-length singularity instead of a finite power-law $\nu$. Scaling theory relates observables through a small number of fixed-point dimensions; it does not make analytic backgrounds, marginal logarithms or dangerously irrelevant variables disappear.

The uniform stationarity equation and local stability condition areFor $u>0$, $h=0$, and small negative $r$, the stable nonzero solution satisfies $M^2=-r/u+O(r^2)$. Thus the order parameter vanishes continuously as $r\uparrow 0$. The equilibrium Landau free energy has leading singular contribution $f_{\min }=-r^2/(4u)$ below the transition and zero above it. Its first thermal derivative is continuous, while its second derivative has a finite jump: the mean-field approximation predicts a continuous phase transition with no latent heat. The magnetic susceptibility follows by differentiating the stationarity equation:It diverges on approaching the ordinary thermodynamic critical point, and the quadratic Landau scalar correlation length behaves as $\xi \sim \sqrt{\kappa /|r|}$, with a different amplitude on the two sides.

For $u<0$, a nonzero stationary point at $h=0$ has $r+u{M}^2+v{M}^4=0$. Substitution into $V(M)-V(0)$ gives $-u{M}^4/4-v{M}^6/3$. Setting this to zero yieldsThe finite jump of $M$ makes this a first-order phase transition. For a thermal path $r=r(T)$ with $r^{\prime }(T)\ne 0$ and the other coefficients fixed, $\partial f_{\min }/\partial T=r^{\prime }(T)M^2/2$ jumps, giving an entropy discontinuity and nonzero latent heat. The Landau free energy itself remains continuous. The disordered minimum loses stability at $r=0$; the two nonzero stationary solutions merge at $r=u^2/(4v)$. These spinodal points bound metastability, but neither replaces the equilibrium phase coexistence condition.

Writing $r=at$ with $a>0$ near an ordinary thermodynamic critical point, the zero-field equation gives $M\sim (-t{)}^{1/2}$. At $r=0$, small $h$ obeys $h=u{M}^3+O(M^5)$, hence $M\sim \mathrm{sgn}(h)|h{|}^{1/3}$. The ordinary mean-field critical exponents are therefore $\beta =1/2$ and $\delta =3$. These are Landau theory predictions, not dimension-independent values for an interacting theory below its upper critical dimension.

Changing the ultraviolet cutoff changes which fluctuations remain explicit. Integrating out a shell of Fourier modes shifts the coefficients of all symmetry-allowed operators and a field-independent constant. Hence the cutoff-dependent coefficients are chosen so that the remaining partition function and long-distance observables reproduce the same microscopic system. This is the Wilsonian coarse-grained statistical Hamiltonian; retaining only the printed operators is a local derivative expansion, not an exact closure.

In a massive symmetric phase, the quadratic long-wavelength inverse two-point function has the form ${\alpha }^{-1}{p}^2+m^2$. Its Landau scalar correlation length satisfies ${\xi }^{-2}=\alpha m^2$ in this approximation. For canonical normalization of a scalar gradient term, set $\varphi =\sqrt{\alpha }\psi$; then the mass and quartic coefficients become $r=\alpha m^2$ and $g_c={\alpha }^{2}g$. With this normalization the infrared mass is the inverse correlation length, $m_R={\xi }^{-1}$, within the massive quadratic or one-loop approximation. More generally the decay rate is a pole mass, whereas the zero-momentum inverse magnetic susceptibility is a curvature mass; an anomalous dimension or nontrivial momentum dependence can distinguish them. In the ordered phase one must first expand about a selected stable equilibrium magnetization, rather than identify a negative quadratic coefficient at the unstable origin with a squared inverse correlation length. Thus the intended mass identification includes a field normalization and a stable-phase qualification.

For momentum-shell renormalization group, split $\varphi ={\varphi }_{<}+{\varphi }_{>}$, retaining $|p|<\Lambda /b$ and eliminating $\Lambda /b<|p|<\Lambda$. DefineTaking a logarithm gives the cumulant expansion of a coarse-grained free energy. For a centered fast Gaussian measure with $G_{>}(0)=⟨{\varphi }_{>}^2⟩$, the first quartic cumulant isThe Wick contractions give a mass shift $\Delta r=g{G}_{>}(0)/2$ and an identity-operator contribution, while leaving the quartic term at this order. The second cumulant produces higher interactions and a quartic correction. After the length and field rescalings $x=b{x}^{\prime }$ and ${\varphi }_{<}(x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$, the gradient coefficient remains one, andThe one-loop shell mass renormalization in scalar quartic theory explicitly shows why the critical bare mass is shifted by fluctuations.

Assume short-range interactions, an analytic even local potential, a positive gradient coefficient, and a weak-coupling trajectory in the basin of the Gaussian fixed point. Above four dimensions the quartic interaction has negative engineering dimension and its dimensionless strength decreases under coarse graining; higher local powers and higher derivative terms also decrease. After absorbing analytic short-distance shifts into the coefficients, long-wavelength fluctuations are small compared with the ordered order parameter. The remaining coarse Landau free energy can therefore be evaluated at its stable saddle:The quartic stabilizer must still be retained for the ordered minimum even though it is dangerously irrelevant. Stationarity, $r_{R}M+g_R{M}^3/6=h$, gives the ordinary Landau-Ginzburg theory and its mean-field critical exponents. This derives the saddle approximation from decreasing fluctuation strength under the stated RG assumptions; it does not assume that every exact coarse-grained functional becomes a quartic polynomial. Below four dimensions the quartic coupling grows, long-wavelength loop corrections cease to be small, and the argument fails. At four dimensions the interaction is marginal and generates logarithmic corrections. The tuned sextic theory has the analogous boundary at three dimensions.

Use a consistent Fourier transform pair for the connected correlation function:The printed forward-transform expression integrates over $p$ while leaving $x$ free; that integration variable is an error. The above convention resolves it and produces the loop measure used below. At zero magnetization, $G_c(x)=⟨\varphi (0)\varphi (x)⟩$; otherwise the disconnected product of expectations must be subtracted.

In the inverse-propagator identity used here, the truncated two-point function means the full one-particle-irreducible two-point vertex, including its free quadratic part. This convention differs from using “truncated” for a connected cumulant. To derive the inverse relation, introduce $W[J]=\log Z[J]$, $\phi =\delta W/\delta J$ and the effective action $\Gamma [\phi ]=\int J\phi -W[J]$. Then $\delta \Gamma /\delta \phi =J$ and the chain rule gives ${\Gamma }^{(2)}{W}^{(2)}=1$. Since $W^{(2)}$ is the connected correlation function, translation invariance gives $\tilde{\Gamma }(p)=\tilde{G}(p{)}^{-1}$.

Now use canonically normalized coefficients, denoting the cutoff mass by $m_b^2=m^2(\Lambda ,T)$ and the renormalized infrared mass by $m_R^2=m^2(0,T)$. Choose the free propagator ${\tilde{G}}_0(p)=(p^2+m_R^2{)}^{-1}$ and put $\delta m^2=m_b^2-m_R^2$ in the interaction as a quadratic counterterm. With the Euclidean self-energy convention in which an insertion changes the connected propagator by $-G_0\Sigma G_0$, the perturbative series beginsThe self-energy sums one-particle-irreducible loop insertions; the explicit mass counterterm is kept separate. Repeated insertions generate the geometric Dyson resummation, which explains why reducible chains occur in $G$ but not as independent terms in its inverse. Beyond one loop a momentum-dependent self-energy and wave-function renormalization must also be included.

At one loop the only quartic two-point graph is the tadpole diagram. At a $g{\varphi }^4/4!$ vertex, attaching the two labeled external fields uses $4\cdot 3=12$ choices and contracts the remaining two fields together. The factor is $12/4!=1/2$. Equivalently this is the same quadratic term in the shell cumulant expansion above. ThusIt is positive for $g>0$ and independent of external momentum, so there is no one-loop gradient renormalization. The condition $\tilde{\Gamma }(0)=m_R^2$ implies $\delta m^2+\Sigma (0)=0$, henceThe cutoff is necessary: the unregulated integral is ultraviolet divergent for $D\ge 2$. Using the renormalized mass internally is a self-consistent one-loop organization, not an exact resummation of all critical fluctuations. With nonunit $\alpha$, this formula instead applies to $r=\alpha m^2$, $g_c={\alpha }^{2}g$ after canonical normalization of a scalar gradient term.

To test a linear thermal mass, approach the ordinary transition from the symmetric side and set $R=m_R^2>0$. For $D>2$ the massless tadpole is infrared finite. Subtract its critical value, absorb regular temperature dependence of the coupling into a nonzero thermal coefficient $A$, and write $m_b^2(T)-m_b^2(T_C)=A(T-T_C)+\cdots$. The one-loop critical-mass subtraction giveswhereFor $D>4$, $J_D(0)=c_D{\Lambda }^{D-4}/(D-4)$ is finite, so the bracket tends to a finite constant and $R\propto T-T_C$ is self-consistent. At $D=4$,and the growing logarithm obstructs an asymptotically constant linear coefficient at nonzero coupling. For $2<D<4$, put $k=\sqrt{R}q$ to obtainThe divergent bracket then invalidates the assumed linear thermal mass. The radial critical-mass subtraction in dimensions three to five gives explicit checks of all three behaviors. For $D\le 2$, the massless tadpole subtraction itself is infrared divergent; that failure of the Gaussian expansion does not prove absence of an interacting transition. Therefore the ordinary upper critical dimension is $D_C=4$, with the unmodified Landau-Ginzburg theory asymptotically justified by this test only for $D>4$. Exponents inferred by solving the self-consistent one-loop equation below four dimensions would not be exact interacting exponents.

For a tricritical point the renormalized quartic coefficient must be tuned to zero, and a positive sextic interaction stabilizes the order parameter. The gradient term gives the scalar field engineering dimension $(D-2)/2$. A coefficient of ${\varphi }^6$ therefore has dimension $D-6(D-2)/2=6-2D$, which becomes marginal at $D=3$. This gives $D_C^{\mathrm{tri}}=3$. A direct Ginzburg criterion reaches the same result: long-wavelength variance in a correlation volume iswhile the tricritical saddle has $M^2\sim |r{|}^{1/2}$. Their ratio scales as $|r{|}^{(D-3)/2}$, vanishing only above three dimensions. At three dimensions it is marginal; below three it grows. In contrast the ordinary saddle $M^2\sim |r|/g$ gives a ratio $g|r{|}^{(D-4)/2}$. Both calculations assume the thermal mass is measured relative to its shifted critical value. A sextic interaction generates quartic terms under coarse graining, so tricriticality requires tuning the renormalized quartic scaling field as well as the mass.

Critical exponents describe leading singular powers near a continuous phase transition. For reduced temperature $t=(T-T_c)/T_c$, common definitions are the spontaneous order parameter $M\sim (-t{)}^{\beta }$, magnetic susceptibility $\chi \sim |t{|}^{-\gamma }$, singular heat capacity $c_s\sim |t{|}^{-\alpha }$, correlation length $\xi \sim |t{|}^{-\nu }$, and the critical equation of state $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. At criticality the connected correlation function has decay power $D-2+\eta$.

In Landau theory, solve the stationary equation, evaluate its stable minimum, and differentiate with respect to the conjugate variables. For a scalar quartic theory with $r=r_{0}t$, $r_0>0$, the equation of state is $h=rM+u{M}^3$. Its ordered solution and differentiated response give $\beta =1/2$ and $\gamma =1$, calculated explicitly in the unheaded solution below. At $r=0$ the same equation gives $\delta =3$. The minimized zero-field singular potential is zero above the transition and $-r^2/(4u)$ below, giving a heat-capacity jump and $\alpha =0$. Expanding the gradient functional about a stable minimum gives inverse quadratic response $\kappa q^2+U^{\prime \prime }(M)$, so $\xi =\sqrt{\kappa /U^{\prime \prime }(M)}\sim |t{|}^{-1/2}$ and the Gaussian critical decay has $\eta =0$.

These are mean-field critical exponents. Renormalization group methods obtain fluctuation-corrected powers from fixed-point scaling dimensions; they need not equal the Landau values below the upper critical dimension. At a marginal dimension logarithms can supplement the leading powers.

A tricritical point occurs when a first-order boundary meets a continuous-transition boundary. In the spin-inversion-symmetric sextic Landau free energy, two independent parameters must tune $r=u=0$, while $v>0$ keeps the potential stable. In the $h=0$ plane, the continuous boundary is $r=0,u>0$, and the first-order boundary is $r=3u^2/(16v),u<0$. They meet at the tricritical point; the order-parameter jump on the first-order side tends to zero as $u\uparrow 0$. The first panel shows this phase diagram, including the metastability limits derived above.

Along the tuned line $u=0,h=0$, the equation of state is $M(r+v{M}^4)=0$. Thus $M=(-r/v{)}^{1/4}$ on a selected ordered branch, rather than the ordinary square-root behavior. Differentiation gives inverse magnetic susceptibility $r$ above and $4|r|$ below, while at criticality $h=v{M}^5$. Hence the mean-field tricritical exponent calculation yields $\beta =1/4$, $\gamma =1$, $\delta =5$; the minimized potential $-|r{|}^{3/2}/(3\sqrt{v})$ also gives $\alpha =1/2$. These tricritical powers require tuning the quartic coefficient, not merely varying a generic temperature path. The ordinary scalar exponents requested separately are calculated next.

For the requested ordinary critical point take $u>0$, $r=r_{0}t$, and the quartic Landau free energy $U(M)=r{M}^2/2+u{M}^4/4-hM$. At $h=0,t<0$, minimization gives $M^2=-r/u$. Choosing the positive pure thermodynamic phase,For the magnetic susceptibility, differentiate the equation of state at fixed temperature:Both sides diverge as $|t{|}^{-1}$, with different amplitudes, soThe one-sided field limit below the transition selects a pure phase; differentiating the symmetric finite-volume mixture would instead include its macroscopic switching response.

In a first-order phase transition, two distinct global minima of the Landau free energy have equal values and exchange stability. The equilibrium order parameter jumps, and a first derivative of the equilibrium free energy is discontinuous. Depending on the direction through the phase diagram, that derivative can be magnetization, entropy or another conjugate quantity; a first-order transition need not have latent heat along every chosen path.

In a continuous phase transition, the equilibrium order parameter changes continuously while the curvature at the critical minimum vanishes. For $u>0$ and $h=0$, $M=0$ is stable for $r>0$, but for $r<0$ the stable minima have $M\simeq \pm \sqrt{-r/u}$. The order parameter therefore goes continuously to zero at $r=0$; the inverse magnetic susceptibility vanishes there.

For $u<0$, the sextic term stabilizes the Landau free energy. The nonzero minima can become favorable while $M=0$ still has positive curvature. At $r=3u^2/(16v)>0$ they coexist with $M=0$, so the order parameter jumps to magnitude $\sqrt{-3u/(4v)}$. Finding a stationary point alone does not locate the transition: equilibrium requires comparing the values of all competing minima. Spinodal boundaries describe loss of local stability, not coexistence.

Write $\beta =1/(k_{B}T)$, with $k_B$ the Boltzmann constant. For independent Ising spins, the one-site partition functions on the two sublattices are $z_B=2\cosh \beta (h+g)$ and $z_W=2\cosh \beta (h-g)$. Differentiating each logarithm with respect to its local field givesAt $h=0$, the two sublattices experience opposite fields, so spin reversal combined with sublattice exchange leaves the Hamiltonian invariant and gives $M_W=-M_B$.

For the antiferromagnetic Ising model, take $N$ sites, equal sublattice sizes and $q=2D$ neighbors per site. There are $Nq/2$ bonds counted once. The two-sublattice Ising mean-field approximation expands each bond asand drops only the product of fluctuations. Summing the remaining terms givesThe printed constant omits $N$; for a total Hamiltonian with unnormalized spin sums, this factor is necessary. Applying the independent-spin calculation now givesThe mean-field approximation is favored by large coordination number: averaging many neighbors reduces local relative fluctuations. The size of critical fluctuations still has to satisfy the Ginzburg criterion; high dimension is a reason for validity, not permission to discard every correlation in every regime.

Antisymmetric solutions and the transition. At $h=0$, putting $M_B=-M_W=M$ makes both equations reduce to $M=\tanh \beta (g+qJM)$. A zero-field continuous transition concerns $g=0$ as well: nonzero $g$ explicitly selects the staggered sign and rounds this critical bifurcation. Put $K=\beta qJ$. The graphical intersections of $M$ and $\tanh (KM)$ give only $M=0$ for $K\le 1$. For $K>1$ there are additionally two nonzero intersections $\pm M_{\ast }$: on $M>0$, $\tanh (KM)-M$ is strictly concave, initially increases, and is negative at $M=1$, so there is exactly one positive root. Spin reversal supplies the negative root. The origin becomes unstable in the staggered direction, whereas the nonzero roots have $K(1-M_{\ast }^2)<1$ and are stable. Expanding about the bifurcation givesThus $M_{-}=(M_B-M_W)/2$ turns on continuously, with $k_B{T}_c=qJ$ and order-parameter critical exponent $1/2$.