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At the tricritical point, both the quadratic and quartic even coefficients vanish, while the sextic coefficient remains positive. Two even control parameters must therefore be tuned, in addition to setting the symmetry-breaking conjugate field to zero. The continuous phase transition line and first-order phase transition line meet, the coexistence order parameter jump shrinks to zero, and the tricritical wings end there. The balance governing the equation of state changes from quartic to sextic stabilization.

Along the tricritical thermal path $u=0$, $r=at$, $h=0$, minimizing $V$ gives $M^4=-r/v$ on the ordered side. At $r=u=0$, the equation of state is $h=v{M}^5$. ConsequentlyThese mean-field critical exponents require the tuned tricritical path; a path with fixed $u>0$ instead has the ordinary exponents derived above. The corresponding equilibrium Landau free energy is $f_{\min }=-(-r{)}^{3/2}/(3\sqrt{v})$ below the transition. Its first thermal derivative is continuous but its second derivative diverges, giving the tricritical mean-field approximation a heat-capacity exponent $\alpha =1/2$. Fluctuations are negligible asymptotically only above the tricritical upper critical dimension three; at three dimensions logarithmic corrections can accompany these powers.

A phase diagram divides a space of control parameters into equilibrium thermodynamic phases, marking phase coexistence boundaries and thermodynamic critical points. Its dimension counts independently variable controls, not spatial dimensions. An explicit three-dimensional example uses the controls $(r,u,h)$ of a sextic even Landau potential:Here $M$ is a scalar order parameter, $h$ its conjugate field, and $r$ a thermal control. The equilibrium magnetization minimizes $V$ globally. At $h=0$, $u>0$, the boundary $r=0$ is a continuous phase transition. At $h=0$, $u<0$, the boundary $r=3u^2/(16v)$ is a first-order phase transition between $M=0$ and $M=\pm \sqrt{-3u/(4v)}$. The two boundaries meet at the tricritical point $r=u=h=0$. A fixed strictly positive quartic coupling alone cannot produce this tricritical point; the stabilizing sextic term is essential when the quartic coefficient is tuned through zero.

The full phase diagram also contains the $h=0$ phase coexistence sheet between positive and negative ordered phases. Crossing it changes the sign of $M$ discontinuously. For $u<0$, the three-phase line $r=3u^2/(16v)$ is the junction of this sheet and two tricritical wings, on which two unequal same-sign equilibrium magnetizations coexist at nonzero $h$. The tricritical wing critical edges are lines of ordinary thermodynamic critical points; they terminate the first-order sheets and meet at the tricritical point. Away from these boundaries, varying $h$ produces a smooth crossover.

For an explicit construction of the tricritical wings, write their two coexisting order parameters as $M_1=s-d$ and $M_2=s+d$, with $s>0$ and $0<d\le s$. Equality of their Landau free energies and first derivatives givesIndeed, for these controls the tricritical wing coexistence factorization isThe last factor is $(M+2s{)}^2+s^2-d^2\ge 0$, proving that both specified minima are global. The reflected wing has negative $h$ and negative $M$. Letting $d\to 0$ gives the tricritical wing critical edge $u=-10v{s}^2/3$, $r=5v{s}^4$, $h=8v{s}^5/3$; there $V^{\prime \prime }(s)=V^{\prime \prime \prime }(s)=0$ and $V^{\prime \prime \prime \prime }(s)=40v{s}^2>0$. Letting $d=s$ recovers the three-phase line. This supplies the first-order sheets and their continuous boundary curves, rather than only a zero-field section.

A tricritical point requires two independent even coefficients in the Landau free energy to vanish: $r=u=0$, with $v>0$ providing stability. At $h=0$, the ordinary critical line $r=0,u>0$ meets the tricritical three-phase line $r=3u^2/(16v),u<0$. The order parameter discontinuity along the first-order line is $\sqrt{-3u/(4v)}$, which goes to zero at their meeting.

The positive sextic coefficient makes this a different leading local potential from an ordinary quartic critical point. Consequently its mean-field critical exponents, crossover behavior and upper critical dimension differ. There are two relevant even tuning directions near the tricritical point, in addition to the conjugate odd field. The root solution and labelled phase diagram display the three controls and the tricritical wings; varying $h$ can give a first-order reversal within the ordered coexistence sheet, a first-order weak/strong transition on a wing, or a continuous transition at a wing's ordinary critical edge.

A phase diagram partitions a space of thermodynamic controls into regions with distinct equilibrium phases. Its boundaries indicate phase coexistence or continuous phase transitions; a critical point is a termination of coexistence where the distinction between phases disappears. A concrete three-dimensional control space is $(T,\Delta ,h)$ for the Blume–Capel model: a spin can be $0$ or $\pm 1$, the exchange favors aligned nonzero spins, and $\Delta$ controls the energetic cost of nonzero spins. Varying $\Delta$ changes the balance between a continuous ordering transition and a first-order phase transition. Their meeting is a tricritical point.

Locally, the same structure is displayed by the Landau free energywith three controls $(r,u,h)$. For $u>0$, the line $r=h=0$ consists of ordinary critical points. For $u<0$, equal free energies of $M=0$ and a nonzero stationary point requireEliminating $r$ gives the tricritical three-phase lineHere $M=0$ and both signs of $M$ coexist, with a discontinuous order parameter. Below this line, or below $r=0$ when $u>0$, the plane $h=0$ is an ordered phase coexistence sheet: crossing it reverses the sign of $M$. Two tricritical wings also extend to nonzero $h$, separating weakly and strongly magnetized states of the same sign. Their ordinary critical points satisfy $f^{\prime }=f^{\prime \prime }=f^{\prime \prime \prime }=0$. Since $f^{\prime \prime \prime }=6uM+20v{M}^3$, their tricritical wing critical edges areAt these edges $f^{\prime \prime \prime \prime }=-12u>0$, giving an ordinary quartic critical expansion around $M_c$. All these structures end at the tricritical point $r=u=h=0$. The diagrams show both the zero-field section and the genuinely three-dimensional coexistence structure.

For the final critical exponent calculation, take $r=at$ with $a>0$, $t=(T-T_c)/T_c$, and hold the other controls fixed. At an ordinary critical point, the quartic term dominates the sextic term. For $h=0$ and $r<0$,whereas on the disordered side the singular contribution is zero. Two temperature derivatives give a finite jump in heat capacity, so $\alpha =0$. At $r=0$, the equation of state is $h=u{M}^3+O(M^5)$, so $\delta =3$.

Along the tricritical trajectory $u=0$, the nonzero minimum obeys $M^4=-r/v$ andThus two temperature derivatives produce $|t{|}^{-1/2}$. At $r=u=0$, $h=v{M}^5$. Therefore the mean-field critical exponents areWith $A$ the thermodynamic free energy, the physical sign is $C=-T{\partial }_T^{2}A$, rather than the positive sign printed in the paper. This correction affects the sign of the heat capacity, not its critical exponent. The tricritical result assumes that the independent quartic control is tuned to zero; a generic trajectory through nearby ordinary critical points need not have these exponents.

At a tricritical point, both $r$ and $u$ vanish, while $v>0$ stabilizes the Landau free energy. It separates ordinary continuous transitions for $u>0$ from first-order coexistence for $u<0$. The zero-field order-parameter jump shrinks to zero as $u\uparrow 0$, and the tricritical wings terminate there in the full $(r,u,h)$ diagram. For $r<0$, $u=0$, minimizing gives $M^4=-r/v$ andTwo thermal derivatives give the heat-capacity critical exponent $\alpha =1/2$, compared with the finite mean-field heat-capacity jump, $\alpha =0$, at an ordinary quartic transition. The remaining tricritical mean-field critical exponents are $(\beta ,\gamma ,\delta )=(1/4,1,5)$, and the mean-field correlation-length critical exponent is $\nu =1/2$. To observe tricritical scaling one must tune the quartic scaling field as well as the thermal field: a generic path with fixed $u>0$ instead has ordinary critical behavior. Its upper critical dimension is three, as the multicritical Ginzburg ratio shows.

A phase diagram locates stable pure thermodynamic phases and their phase coexistence boundaries in a space of control parameters. For a concrete three-dimensional example, take the temperature, crystal-field parameter, and magnetic field of the Blume–Capel model. Its spins take values $0,\pm 1$; changing the cost of the zero-spin state can turn a continuous ordering transition into a discontinuous one. A local scalar description near the resulting tricritical point is the Landau-Ginzburg theoryThe three local control coordinates are $(r,u,h)$, with $r$ varying linearly with the thermal scaling field. For $u>0$, $h=0$, the transition at $r=0$ is continuous. For $u<0$, zero-field coexistence is at $r=3u^2/(16v)$, and the magnetization jumps between zero and $\pm \sqrt{-3u/(4v)}$. The two loci meet at $r=u=h=0$, the tricritical point.

The full three-dimensional phase diagram has an ordered coexistence sheet at $h=0$, where the two opposite magnetizations exchange stability when $h$ changes sign. On its three-phase line it meets two nonzero-field tricritical wings. These wings are first-order phase transition surfaces between minima of the same sign but different magnitude, terminating at ordinary thermodynamic critical points. Their tricritical wing critical edges are found by $V^{\prime }=V^{\prime \prime }=V^{\prime \prime \prime }=0$:Indeed $V^{\prime \prime \prime }=6uM+20v{M}^3$ fixes $M_c^2$, and substituting into $V^{\prime \prime }=0$ and $V^{\prime }=0$ gives $r_c,h_c$; $V^{\prime \prime \prime \prime }=-12u>0$ ensures a stable critical quartic minimum. The two signs give the two wings. All transition surfaces and critical edges end together at the tricritical point. The following plot uses $v=1$; its wing surfaces come from the exact tricritical wing coexistence factorization, rather than schematic unconstrained surfaces.

The unheaded requests for critical exponents follow from the uniform equation of stateFor an ordinary thermodynamic critical point, $u>0$ is fixed and $r=At$, $A>0$. At $h=0$, $t<0$, the small ordered minimum obeys $M^2\sim -r/u$, so the order-parameter critical exponent is $\beta =1/2$. Above the transition $\chi =(\partial M/\partial h{)}_{h=0}=1/r$, giving the magnetic-susceptibility critical exponent $\gamma =1$. At $r=0$, $h\sim u{M}^3$, so the critical-isotherm exponent is $\delta =3$. At the tuned tricritical point, $u=0$, instead $M^4=-r/v$ below the transition and $h=v{M}^5$ on its critical isotherm, while the disordered susceptibility remains $1/r$. ThereforeBelow the transition the inverse susceptibility is $r+3u{M}^2+5v{M}^4$, giving $2|r|$ in the ordinary limit and $4|r|$ in the tricritical limit. Thus both sides give the same susceptibility power, with different amplitudes.

To derive the general Ginzburg criterion, tune away all lower even nonlinearities and let the leading stabilizer be $v_n{M}^{2n}/(2n)$, $n\ge 2$. The equation of state gives $M_0^2\asymp |r{|}^{1/(n-1)}$, and its curvature at the ordered saddle is $2(n-1)|r|$, so $\xi \asymp |r{|}^{-1/2}$. The fluctuations of the order parameter averaged over a correlation volume are estimated by its long-wavelength quadratic covariance:The integral has a fixed dimensionless upper limit after $q=p/\xi$; this is a coarse-volume fluctuation estimate, not the ultraviolet-dominated point variance. Consequently the multicritical Ginzburg ratio isThe Landau approximation is asymptotically controlled by this test only when the ratio tends to zero. This gives the upper critical dimensionThe same threshold follows by giving $M$ its Gaussian engineering dimension $(D-2)/2$: the coupling to $M^{2n}$ has scaling exponent $D-n(D-2)=2n-(n-1)D$, which becomes relevant below $D_c$.

For $D<D_c$, the fluctuation ratio diverges and the saddle calculation cannot predict the generic interacting critical exponents. At $D=D_c$ the test is marginal, not a proof of changed power exponents: the ratio has zero thermal power and marginal interactions can produce logarithmic corrections. Thus the PDF's inclusion of equality in a blanket claim about wrong power exponents needs this qualification. The Landau-Ginzburg theory as a fluctuating field theory remains useful; what loses validity is its fluctuation-free Landau approximation.

A phase diagram partitions a space of control parameters into regions with distinct equilibrium phases. Its boundaries mark nonanalyticities of the thermodynamic-limit free energy; a first-order phase transition has a jump in a first derivative, whereas a continuous phase transition has a continuously vanishing order parameter and singular fluctuations.

An explicit three-parameter example is the scalar Landau free energywith parameter axes $(r,u,h)$. These can be realized locally by the temperature, single-ion anisotropy and magnetic field of the Blume–Capel model. The three dimensions here refer to control-parameter space, not necessarily to three spatial dimensions. At $h=0$, minimizing this potential gives a continuous phase transition at $r=0$ for $u>0$. For $u<0$, coexistence occurs at $r=3u^2/(16v)$, whereThere are three global minima, $M=0$ and $M=\pm \sqrt{-3u/(4v)}$. Their order parameter jump decreases to zero as $u\uparrow 0$. Thus $(r,u,h)=(0,0,0)$ is a tricritical point. The two transition curves in the $h=0$ plane meet there.

In the three-dimensional phase diagram, crossing $h=0$ inside the ordered region switches between opposite signs of $M$, so that region is itself a first-order phase transition sheet. For $u<0$, two tricritical wings at opposite nonzero fields separate weakly and strongly magnetized phases. Their wing critical edges can be obtained rather than merely sketched: require $V^{\prime }=V^{\prime \prime }=V^{\prime \prime \prime }=0$. Since $V^{\prime \prime \prime }=6uM+20v{M}^3$, the nonzero solutions satisfyThe fourth derivative there is $-12u>0$, giving ordinary continuous critical endpoints. The wings meet the zero-field coexistence sheet along the tricritical three-phase line and terminate at these edges. Their exact tricritical wing coexistence factorization supplies the surfaces in the figure: for two nonnegative minima $a\le b$, set $s=a+b$, $p=ab$ andDirect expansion gives $V(M)-V(a)=(v/6)(M-a{)}^2(M-b{)}^2[(M+s{)}^2+p]\ge 0$, proving that these surfaces describe coexistence of global, not merely local, minima.

For the unheaded Ising model calculation, assume ferromagnetic exchange $J>0$ and let $M$ denote the mean Ising spin. On the hypercubic lattice the coordination number of a lattice is $z=2D$, even though the original positive-direction bond sum counts each bond only once. The mean-field approximation neglects products of deviations from $M$:There are $ND$ bonds, so the approximate energy is $H_{\mathrm{MF}}=-2DJM{\sum }_i{\sigma }_i+NDJ{M}^2$. Each independent Ising spin therefore sees the effective field $2DJM$. With ${\beta }_{\mathrm{th}}=1/(k_{B}T)$, averaging its two Boltzmann weights gives the self-consistency equationThe slope of its right side at zero is $T_c/T$, whereAbove $T_c$, $\tanh x<x$ for $x>0$ excludes a nonzero solution. Below $T_c$ the initial slope exceeds one, and concavity of the hyperbolic tangent on the positive axis gives one positive solution and its negative partner. The Ising auxiliary mean-field free energy $DJ{M}^2-k_{B}T\log [2\cosh (2{\beta }_{\mathrm{th}}DJM)]$ has positive quartic coefficient near $T_c$, while its quadratic coefficient changes sign there. Its stable minima therefore move continuously away from zero; this is a continuous phase transition in the mean-field theory of the Ising model.

Put $\tau =(T_c-T)/T_c>0$ and $c=T_c/T=(1-\tau {)}^{-1}$. The Taylor expansion $\tanh x=x-x^3/3+O(x^5)$ gives, on the nonzero branch,Consequently $M\sim \sqrt{3}{\tau }^{1/2}$ and the mean-field order-parameter exponent is $\beta =1/2$. The inverse-temperature symbol ${\beta }_{\mathrm{th}}$ used above is distinct from this order-parameter critical exponent.

A tricritical point occurs at $r=u=h=0$ with $v>0$: tuning $u$ changes a continuous transition into a first-order one. At $u=0$, the equation of state is $h=rm+v{m}^5$. It gives $m\propto (-r{)}^{1/4}$ at $h=0$, $m\propto h^{1/5}$ at $r=0$, and a minimized free energy proportional to $-|r{|}^{3/2}$ below the transition. The mean-field tricritical exponents areIn the three-dimensional parameter space $(r,u,h)$, the zero-field disordered-to-ordered continuous line $r=0,u>0$ meets the first-order line $r=3u^2/(16v),u<0$ at the tricritical point. The zero-field surface also contains coexistence of positive and negative ordered phases. For negative $u$, tilting with $h$ produces two symmetric tricritical wings of first-order coexistence, each bounded by a line of ordinary critical endpoints. Those wings and their critical edges meet at the tricritical point. The upper critical dimension of tricritical sextic theory is three, so mean-field values can acquire logarithmic corrections there.

Consider the $(u,r)$ plane at $h=0$ for $V=r{M}^2/2+u{M}^4/4+v{M}^6/6$, with fixed $v>0$. For $u>0$, the continuous boundary is $r=0$: above it the global minimum is zero, and below it there are two symmetry-related ordered pure thermodynamic phases. For $u<0$, the equilibrium boundary is the tricritical three-phase lineAbove this curve the disordered minimum wins; below it the two ordered minima win. On the curve,Its three zeros are exactly $M=0$ and $M=\pm \sqrt{-3u/(4v)}$, proving equality of three global minimum values. Spin inversion symmetry enforces the equality of the two ordered phases automatically, so tuning a single remaining coefficient can produce three-phase coexistence on a line in this symmetry plane. This does not contradict the generic phase-rule counting without symmetry restrictions.

The tricritical point is $(u,r)=(0,0)$, where the discontinuity vanishes, the first-order boundary meets the continuous boundary, and the positive sextic term becomes the leading stabilizer. The local metastability limits $r=0$ and $r=u^2/(4v)$, $u<0$, bracket the first-order curve but are not phase boundaries. In the full $(r,u,h)$ parameter space, the zero-field line is attached to tricritical wings; a nonzero field splits the equality of the two ordered phases.

A tricritical point joins continuous and first-order transition loci and requires tuning an additional temperature-independent control. In the scalar sextic potentialit occurs at $r=u=h=0$: both the quadratic and quartic terms vanish. At $h=0,u>0$, the transition is continuous at $r=0$; at $h=0,u<0$, it is first-order at $r=3u^2/(16v)$. The jump tends to zero as $u\uparrow 0$, so these loci meet at the tricritical point.

For the three-control phase diagram with coordinates $(u,r,h)$, the sheet $h=0$ below these transition loci has coexistence of the two ordered phases. For $u>0$ its boundary is the ordinary continuous critical line $r=h=0$. For $u<0$ the boundary $r=3u^2/(16v),h=0$ is a tricritical three-phase line, where $m=0$ and $m=\pm \sqrt{-3u/(4v)}$ all coexist. Two symmetry-related tricritical wings extend from that line into $h>0$ and $h<0$, describing first-order coexistence between small- and large-magnitude phases of the same sign. Each wing ends along an ordinary wing critical edge.

The critical-edge conditions $V^{\prime }=V^{\prime \prime }=V^{(3)}=0$ giveHere $V^{(4)}(m_c)=-12u>0$, so these are ordinary quartic endpoints. The two wing critical edges and the zero-field ordinary critical line are three continuous critical lines meeting at the tricritical point; the three-phase first-order line also ends there. The “3D” phase diagram refers to three independent controls; it is distinct from the spatial dimension $D$ of the field theory.

The figure uses actual sextic-potential coexistence surfaces rather than drawing arbitrary wings. For two coexisting nonnegative minima $a\le b$, put $s=a+b,q=ab$. The tricritical wing coexistence factorizationholds when $u=(2v/3)(-2s^2+3q)$, $r=(v/3)(s^4-s^{2}q+3q^2)$ and $h=(v/3)s^{3}q$. It proves that both minima are globally stable. The limits $q=0$ and $q=s^2/4$ produce the three-phase line and critical edge, respectively; reflection gives the other wing.

At $u=h=0$, $m_0=(-r/v{)}^{1/4}$ on the ordered side and $V(m_0)=-(-r{)}^{3/2}/(3\sqrt{v})$. At the critical isotherm $h=v{m}^5$. Thus the tricritical mean-field critical exponents areThe source's extra control is required to tune $u$ to zero; simply changing temperature in a generic quartic system does not produce tricriticality. The tricritical upper critical dimension is three, as established in Question 3.

For $V=r{m}^2/2+u{m}^4/4+v{m}^6/6-hm$ with $v>0$, the displayed line has three global minima, $m=0$ and $m=\pm \sqrt{-3u/(4v)}$. The factorization $V=(v/6)m^2(m^2+3u/(4v){)}^2$ proves their coexistence. Two nonzero-field tricritical wings and the zero-field ordered coexistence sheet meet along this line. Its order-parameter discontinuity tends to zero as it terminates at the tricritical point.

For $f=r{M}^2/2+u{M}^4/4+v{M}^6/6-hM$ with $v>0$, the ordinary critical edge of a tricritical wing satisfies $f^{(1)}=f^{(2)}=f^{(3)}=0$. For $u<0$, $M_c^2=-3u/(10v)$, $r_c=9u^2/(20v)$ and $h_c=6u^2{M}_c/(25v)$. Its fourth derivative is $-12u>0$. The two signs of $M_c$ give the two wings.