Tricritical Points

Most continuous phase transitions are organised around an ordinary critical point, reached by tuning a single field (temperature) to the boundary of an ordered phase. A tricritical point is a rarer and richer object: it is the special point where a line of continuous (second-order) transitions meets a line of discontinuous (first-order) transitions and terminates. Approaching it, the latent heat and the order-parameter jump of the first-order line shrink to zero exactly as the line joins the critical line, so the transition smoothly changes its very order. Reaching such a point requires tuning two independent fields — it is a codimension-two feature of the phase diagram. Landau theory captures this with a single extra term. An ordinary critical point is described by the symmetric quartic free energy $f=\tfrac12 a m^2+\tfrac14 u m^4$ with $u>0$ . When some second control field drives the quartic coefficient $u$ through zero and negative, the quartic well can no longer bound the free energy, and the next allowed symmetric term $\tfrac16 w m^6$ (with $w>0$ ) must be retained. For $u>0$ the transition stays continuous; for $u<0$ it becomes first order with a jump; the boundary case $u=0$ is the tricritical point. Both conditions $a=0$ and $u=0$ must hold, which is why two fields are needed. Because the sextic term now governs the transition, the mean-field critical exponents change from their ordinary values to the tricritical set $\beta_t=\tfrac14$ , $\delta_t=5$ , $\gamma_t=1$ , $\alpha_t=\tfrac12$ . These satisfy the same scaling relations $\alpha+2\beta+\gamma=2$ and $\gamma=\beta(\delta-1)$ , but with different numbers. Remarkably, they are observed accurately in three-dimensional experiments, because power counting on the $\phi^6$ coupling gives an upper critical dimension $d_c=3$ — so real 3D tricritical points sit exactly at their marginal dimension and show mean-field exponents dressed only with logarithmic corrections. Concrete realizations include the superfluid transition in $^3$ He– $^4$ He mixtures, metamagnets in a field, and the Blume–Capel and Blume–Emery–Griffiths spin-1 Ising models introduced to describe exactly such systems. In every case a non-ordering second field (concentration, uniform field, or single-ion anisotropy) plays the role of $g\propto u$ , tuning the transition from continuous to first order across the tricritical point.

Key takeaways

A tricritical point is where a line of continuous transitions meets and terminates a line of first-order transitions; the transition changes order there.
It is codimension two: two independent fields (temperature AND a second field such as concentration or magnetic field) must be tuned, because both $a=0$ and $u=0$ are required.
Landau theory needs the sextic term: $f=\tfrac12 a m^2+\tfrac14 u m^4+\tfrac16 w m^6$ ; $u>0$ gives 2nd order, $u<0$ gives 1st order (jump $m^{*2}=-3u/4w$ at $a^*=3u^2/16w$ ), $u=0$ is tricritical.
The mean-field tricritical exponents are $\beta_t=\tfrac14,\ \delta_t=5,\ \gamma_t=1,\ \alpha_t=\tfrac12$ , obeying $\alpha+2\beta+\gamma=2$ and $\gamma=\beta(\delta-1)$ .
The upper critical dimension is $d_c=3$ (the $\phi^6$ coupling is marginal there), so real 3D systems show mean-field tricritical exponents with only logarithmic corrections.
Key idea: When the Landau quartic coefficient is tuned to zero, the sextic term takes over; the point $a=u=0$ where the first-order and second-order lines join is the tricritical point, with mean-field exponents $\beta_t=\tfrac14,\delta_t=5$ exact for $d\ge d_c=3$ .

The core insight

When the Landau quartic coefficient $u$ is tuned to zero, the sextic term $\tfrac16 w m^6$ governs the transition; the point $a=u=0$ where the first-order and second-order lines meet is the tricritical point, with mean-field exponents $\beta_t=\tfrac14,\ \delta_t=5,\ \gamma_t=1,\ \alpha_t=\tfrac12$ that are exact above the upper critical dimension $d_c=3$ .

Where this lives today

Tricritical points are studied wherever a phase boundary changes from continuous to first order under a second control field. The classic laboratory is the $^3$ He– $^4$ He mixture, where the superfluid $\lambda$ -line meets the first-order phase-separation line at a tricritical point near $T_t\approx0.87$ K and $^3$ He concentration $x_t\approx0.67$ ; the two tuning fields are temperature and concentration, and the measured exponents confirm the $d_c=3$ mean-field prediction. Metamagnets such as FeCl $_2$ in a magnetic field show a tricritical point in the $(T,H)$ plane where the antiferromagnetic transition switches from continuous to first order. The Blume–Capel and Blume–Emery–Griffiths spin-1 Ising models — introduced precisely to model these systems — remain workhorses for numerical studies of tricriticality, and the same $\phi^6$ structure appears in the collapse (theta) transition of polymers and in microemulsion phase diagrams, where an effective quartic coupling passes through zero.

The physical picture

Draw the phase diagram in the plane of two control fields — say temperature and concentration. A line of continuous transitions separates ordered from disordered phases; follow it and at some point it stops being continuous and turns into a line of first-order transitions, across which the order parameter jumps and latent heat is released. The junction where the smooth line becomes the jumpy line is the tricritical point. As you slide along the first-order line toward it, the jump and the latent heat shrink to nothing, healing continuously into the critical line. That healing is the defining signature: the order of the transition changes there without any discontinuity in the location of the transition itself.

Figure

Phase diagram — a second-order line meets a first-order line at the tricritical point

The mechanism lives in the Landau free energy. An ordinary transition is the competition between a quadratic well that flips sign at $T_c$ and a stabilizing quartic term. Tune a second field and you can drive the quartic coefficient to zero. Once it goes negative the quartic term itself would send the free energy to $-\infty$ , so the sextic term must step in to stabilize the minimum — and its shape produces a first-order jump. The tricritical point is the knife-edge where the quartic coefficient is exactly zero: the sextic term alone shapes a continuous but softer transition, giving new exponents.

Figure

Landau free energy f(m) — quartic-well continuous vs sextic-driven first-order jump, tricritical in between

The core equations

The tricritical Landau free energy retains the sextic term, needed because the quartic coefficient can vanish and change sign. All the physics of the article follows from minimizing it.

f(m)=\tfrac12 a\,m^2+\tfrac14 u\,m^4+\tfrac16 w\,m^6-h\,m,\qquad w>0 \qquad (1)

where: $m$ is the order parameter; $a\propto t=(T-T_c)/T_c$ the quadratic coefficient; $u$ the quartic coefficient (tuned by a second field); $w>0$ the sextic coefficient ensuring stability; $h$ the ordering field. Physically: $w>0$ bounds $f$ below even when $u\le0$ . The equilibrium condition is the vanishing of the field-free derivative, a quintic in $m$ .

\frac{\partial f}{\partial m}=a\,m+u\,m^3+w\,m^5=h \qquad (2)

where: the roots at $h=0$ are $m=0$ and the ordered branch $m^2=\dfrac{-u+\sqrt{u^2-4aw}}{2w}$ . When the quartic coefficient is negative the transition is first order, occurring at a shifted temperature with a finite jump.

a^*=\frac{3u^2}{16w},\qquad m^{*2}=\frac{-3u}{4w}\quad(u<0) \qquad (3)

where: $a^*$ is the quadratic coefficient at coexistence; $m^*$ the order-parameter jump. Physically: two symmetric ordered minima become degenerate with the $m=0$ state. Setting $u=0$ leaves the pure sextic transition, whose order parameter and critical isotherm define the tricritical exponents $\beta_t$ and $\delta_t$ .

a\,m+w\,m^5=h:\quad m\sim(-a/w)^{1/4}\ \Rightarrow\ \beta_t=\tfrac14,\qquad h\big|_{a=0}=w\,m^5\ \Rightarrow\ \delta_t=5 \qquad (4)

where: $\beta_t=\tfrac14$ is the tricritical order-parameter exponent; $\delta_t=5$ the tricritical-isotherm exponent. The susceptibility and specific heat give the remaining two mean-field tricritical exponents, and the whole set obeys the standard scaling relations.

\gamma_t=1,\quad \alpha_t=\tfrac12,\qquad \alpha_t+2\beta_t+\gamma_t=2,\quad \gamma_t=\beta_t(\delta_t-1) \qquad (5)

where: $\gamma_t$ is the susceptibility exponent, $\alpha_t$ the specific-heat exponent. Physically: the exponents differ from the ordinary set but satisfy the same relations. Tricriticality needs a second relevant field, encoded by a crossover exponent, and its upper critical dimension follows from power counting on the sextic coupling.

f_{\rm sing}=|t|^{2-\alpha_t}\,\Phi\!\left(\frac{h}{|t|^{\Delta_t}},\frac{g}{|t|^{\phi_t}}\right),\ \Delta_t=\beta_t\delta_t=\tfrac54,\ \phi_t=\tfrac12,\qquad [w]=6-2d\ \Rightarrow\ d_c=3 \qquad (6)

where: $g\propto u$ is the non-ordering second field; $\Delta_t=\tfrac54$ the gap exponent; $\phi_t=\tfrac12$ the crossover exponent; $d_c=3$ the upper critical dimension.

Derivation AOrder classification when the quartic coefficient passes through zero

The goal is to show that the transition driven by the quadratic coefficient is continuous when the quartic coefficient is positive but first order when it is negative, with the boundary case defining the tricritical point. We solve the field-free equilibrium condition for the ordered branch, follow the position of the global minimum as the quadratic coefficient decreases at fixed quartic and sextic coefficients, and track the order-parameter jump and coexistence temperature. As the quartic coefficient is sent to zero both the jump and the temperature shift vanish, so the first-order line terminates smoothly on the critical line — that terminal point is the tricritical point, and this single calculation classifies the entire neighbourhood of the phase diagram.

Step 1. Equilibria at $h=0$ : $am+um^3+wm^5=0$ gives $m=0$ and $wm^4+um^2+a=0$ , so the physical ordered branch is $m^2=\frac{-u+\sqrt{u^2-4aw}}{2w}$ . Why: factoring out $m$ leaves a quadratic in $m^2$ ; the $+$ root is the stable minimum.

Step 2. Case $u>0$ : as $a$ decreases through zero, $m^2$ grows continuously from zero like $(-a/u)^{1/2}$ — no jump, a continuous transition. Why: with $u>0$ the ordered minimum detaches from $m=0$ smoothly as the quadratic well flips sign.

Step 3. Case $u<0$ : for small positive $a$ a nontrivial minimum already exists, separated from $m=0$ by a barrier; the two become degenerate at $a^*=3u^2/16w$ , where $m$ jumps by $m^{*2}=-3u/4w$ — a first-order transition. Why: a negative quartic deepens the ordered well early, creating coexistence with $m=0$ .

Step 4. Boundary $u=0$ : both $a^*=3u^2/16w\to0$ and $m^{*2}=-3u/4w\to0$ as $u\to0$ , so the first-order line terminates on the critical line at $a=u=0$ — the tricritical point. Why: the jump and its temperature shift are both proportional to powers of $u$ , healing to zero together. *Verified: verify_aq-sm-tricritical-points.py solves the quintic symbolically, confirms the coexistence condition (equal-depth minima) gives $a^*=3u^2/16w$ and $m^{*2}=-3u/4w$ , and checks the jump and shift vanish as $u\to0$ .*

Go deeper · Derivation B · 1 minMean-field tricritical exponents from the sextic potential

Derivation BMean-field tricritical exponents from the sextic potential

The goal is to derive the four mean-field tricritical exponents directly from the pure sextic free energy obtained at the tricritical point, and to confirm they satisfy the standard scaling relations. We set the quartic coefficient to zero, add a field term, and extract each exponent in turn: the order parameter from the field-free minimum, the critical isotherm from the field response at the transition temperature, the susceptibility from the second derivative above and below, and the specific heat from the temperature dependence of the minimized free energy. The resulting set differs from the ordinary mean-field exponents precisely because the confining term is now sixth order rather than fourth, giving a softer well and a larger response.

Step 1. Order parameter: at $u=0,\ h=0$ , $am+wm^5=0\Rightarrow m=(-a/w)^{1/4}$ , and $a\propto t$ gives $m\sim|t|^{1/4}$ , so $\beta_t=\tfrac14$ . Why: the sextic confinement makes $m$ grow as the fourth root, softer than the ordinary square root.

Step 2. Critical isotherm: at $a=0$ , $h=\partial f/\partial m=wm^5$ , so $h\sim m^5$ and $\delta_t=5$ . Why: on the tricritical isotherm only the sextic term survives, giving a fifth-power response.

Step 3. Susceptibility: $\chi^{-1}=\partial^2 f/\partial m^2=a+5wm^4$ . Above $T_c$ ( $m=0$ ): $\chi^{-1}=a\Rightarrow\chi\sim t^{-1}$ . Below: $m^4=-a/w\Rightarrow\chi^{-1}=a-5a=-4a\Rightarrow\chi\sim|t|^{-1}$ . So $\gamma_t=1$ both sides. Why: the second derivative at the minimum gives the same power above and below.

Step 4. Specific heat: with $m^2=(-a/w)^{1/2}$ , $f_{\min}=-\tfrac13(-a)^{3/2}w^{-1/2}$ , so $C\sim-T\,\partial^2 f/\partial T^2\sim|t|^{-1/2}$ , giving $\alpha_t=\tfrac12$ . Check: $\alpha_t+2\beta_t+\gamma_t=\tfrac12+\tfrac12+1=2$ and $\gamma_t=\beta_t(\delta_t-1)=\tfrac14\cdot4=1$ . Why: substituting the minimum energy and differentiating twice yields the half-power singularity, and the scaling relations hold identically. *Verified: verify_aq-sm-tricritical-points.py derives β_t=¼, δ_t=5, γ_t=1, α_t=½ symbolically from f=½am²+⅙wm⁶ and checks α+2β+γ=2 and γ=β(δ−1).*

Go deeper · Derivation C · 1 minUpper critical dimension d = 3 by power counting on the sextic coupling

Derivation CUpper critical dimension d = 3 by power counting on the sextic coupling

The goal is to show that thermal fluctuations render the sextic tricritical theory marginal at three dimensions, so that mean-field tricritical exponents are exact for $d\ge3$ and dressed only by logarithms in $d=3$ . We assign engineering dimensions to the field from the gradient term of the Landau–Ginzburg–Wilson functional, then require the sextic term to be dimensionless to read off the scaling dimension of its coupling. The coupling turns out to be irrelevant above three dimensions and marginal at exactly three, which is why real three-dimensional tricritical points — unlike ordinary critical points, whose upper critical dimension is four — display their mean-field exponents cleanly.

Step 1. Field dimension: fixing $\tfrac12(\nabla\phi)^2$ to be dimensionless under $x\to bx$ gives $[\phi]=\tfrac{d-2}{2}$ . Why: the gradient carries dimension one, so the field must scale to keep the kinetic term invariant.

Step 2. Sextic coupling: $[w\phi^6]=d\Rightarrow[w]=d-6\cdot\tfrac{d-2}{2}=d-3(d-2)=6-2d=2(3-d)$ . Why: the full term must have dimension $d$ ; note $6\cdot\tfrac{d-2}{2}=3(d-2)=3d-6$ , so $[w]=d-(3d-6)=6-2d$ — a base draft that writes $[w]=3-d$ dropped a factor of two (a cheat-sheet trap below), though it happens to vanish at the same $d=3$ .

Step 3. Read off: $[w]=2(3-d)$ is negative for $d>3$ (irrelevant), zero at $d=3$ (marginal), positive for $d<3$ (relevant); hence $d_c=3$ , and the general $\phi^{2n}$ rule $d_c=2n/(n-1)$ gives $n=3\Rightarrow3$ (and $n=2\Rightarrow4$ for ordinary criticality). Why: a coupling with negative scaling dimension flows to zero, leaving Gaussian/mean-field behaviour. *Verified: verify_aq-sm-tricritical-points.py checks $[w]=6-2d$ (correcting the base's $3-d$ ), that it vanishes at $d=3$ , and the general formula $d_c=2n/(n-1)$ , recovering 3 for the sextic and 4 for the quartic.*

Limiting cases

Behaviour of the sextic Landau functional in regimes of its coefficients $(a,u)$ with $w>0$ fixed:

Case	Result	Meaning
Critical line ( $u>0,\ a\to0^+$ )	$m\sim(-a/u)^{1/2}$ , $\beta=\tfrac12,\ \delta=3$	Ordinary continuous transition; sextic negligible, standard mean-field exponents.
Tricritical point ( $u=0,\ a\to0$ )	$m\sim(-a/w)^{1/4}$ , $\beta_t=\tfrac14,\ \delta_t=5$	Quartic gone; sextic governs; new exponents.
First-order line ( $u<0$ )	jump at $a^=3u^2/16w$ , $m^{2}=-3u/4w$	Two competing minima; discontinuous jump, latent heat $>0$ .
Upper critical dimension	$d_c=2n/(n-1)=3$ ( $n=3$ )	The $\phi^6$ coupling is marginal at $d=3$ ; mean-field tricritical exponents hold with log corrections in $d=3$ .

Worked examples

Example 1 — the first-order jump and where it heals. For the sextic Landau free energy with $u<0$ , compute the coexistence temperature and the order-parameter jump, and show both vanish at the tricritical point.

Step 1. Coexistence requires the ordered minima to be degenerate with $m=0$ : solving $f(m^*)=0$ together with $f'(m^*)=0$ gives $a^*=3u^2/16w$ and $m^{*2}=-3u/4w$ . With, say, $u=-4$ and $w=1$ : $a^*=3\cdot16/16=3$ and $m^{*2}=12/4=3$ , so $m^*=\sqrt3\approx1.732$ .

Step 2. These are the location and size of the first-order transition. Now shrink $u$ toward zero: at $u=-1$ , $a^*=3/16=0.1875$ and $m^{*2}=3/4=0.75$ ; at $u=-0.1$ , $a^*=0.001875$ and $m^{*2}=0.075$ .

Step 3. Both the coexistence shift $a^*\propto u^2$ and the jump $m^{*2}\propto|u|$ collapse to zero as $u\to0$ — the first-order line heals continuously into the critical line, which is exactly what makes $a=u=0$ a tricritical point rather than an isolated first-order endpoint.

Example 2 — checking the tricritical scaling relations numerically. Confirm the mean-field tricritical exponent set satisfies Rushbrooke and Widom, and contrast with the ordinary set.

Step 1. Tricritical set: $\alpha_t=\tfrac12,\ \beta_t=\tfrac14,\ \gamma_t=1,\ \delta_t=5$ . Rushbrooke: $\alpha_t+2\beta_t+\gamma_t=\tfrac12+2\cdot\tfrac14+1=\tfrac12+\tfrac12+1=\mathbf{2}$ ✓.

Step 2. Widom: $\beta_t(\delta_t-1)=\tfrac14\cdot(5-1)=\tfrac14\cdot4=\mathbf{1}=\gamma_t$ ✓.

Step 3. Ordinary set for contrast: $\alpha=0,\ \beta=\tfrac12,\ \gamma=1,\ \delta=3$ : Rushbrooke $0+1+1=2$ ✓, Widom $\tfrac12\cdot2=1$ ✓. Both sets obey the same relations with different numbers — the relations are structural, the exponents are not. The gap exponent shifts from $\Delta=\beta\delta=\tfrac32$ (ordinary) to $\Delta_t=\beta_t\delta_t=\tfrac54$ (tricritical).

Origins · 1 minpierre curie and how this was discovered

pierre curie

1859–1906 · Pierre Curie, working before the modern theory of phase transitions, established the experimental foundation on which tricriticality later rested. In his 1895 doctoral work on magnetism he showed that a ferromagnet loses its spontaneous magnetization at a sharp temperature — the Curie temperature T_c — and formulated Curie's law for the paramagnetic susceptibility above it. This identification of a continuous ordering transition with a well-defined critical temperature is the ancestor of the order-parameter picture: the magnetization is the prototype order parameter, and the ferromagnetic transition is the archetype whose Landau free energy, extended by a sextic term, produces tricritical behaviour. Curie's insistence that the transition be characterized by measurable critical quantities set the template for the exponents β, γ, δ that distinguish ordinary from tricritical points. He shared the 1903 Nobel Prize in Physics with Marie Curie and Henri Becquerel for the discovery of radioactivity.

History

1895

— Pierre Curie identifies the ferromagnetic Curie temperature and Curie's law

1937

— Landau's theory of continuous transitions from a symmetric free energy

1970

— Griffiths coins the term tricritical point and analyses the sextic Landau theory

1971

— Blume, Emery and Griffiths introduce the spin-1 model for the ³He–⁴He mixture

1971

— Riedel and Wegner give the renormalization-group treatment and d_c = 3

1974

— tricritical exponents measured in ³He–⁴He mixtures near T_t ≈ 0.87 K

1984

— Lawrie and Sarbach review tricritical phenomena and confirm the mean-field picture in d = 3

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 12 (Landau theory and tricritical points) · Kardar, Statistical Physics of Fields, Ch. 3–4 · Landau & Lifshitz, Statistical Physics Part 1, §150–152 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 9 · I. D. Lawrie & S. Sarbach, in Phase Transitions and Critical Phenomena vol. 9 (1984) · M. Blume, V. J. Emery, R. B. Griffiths, Phys. Rev. A 4, 1071 (1971).

Provenance. The Landau free energy, equilibrium roots, the first-order line $a^*=3u^2/16w$ with jump $m^{*2}=-3u/4w$ , and the mean-field exponents $(\alpha,\beta,\gamma,\delta)=(\tfrac12,\tfrac14,1,5)$ are standard (Pathria & Beale Ch. 12; L&L §150–152; Kardar). The scaling checks $\alpha+2\beta+\gamma=2$ and $\gamma=\beta(\delta-1)$ hold for the tricritical set, and $d_c=3$ follows from $[w]=6-2d=2(3-d)$ (a base draft's $[w]=3-d$ is off by a factor of two, corrected here) and the general $\phi^{2n}$ rule $d_c=2n/(n-1)$ . The $^3$ He– $^4$ He tricritical coordinates ( $T_t\approx0.87$ K, $x_t\approx0.67$ ) are quoted from standard references and rechecked against the phase-diagram literature.

Builds on: Landau Theory of Phase Transitions · Continuous Phase Transitions · First-Order Phase Transitions · Order Parameter · Critical Exponents. Leads to: Universality and Universality Classes · Scaling Hypothesis and Scaling Relations · Latent Heat and Phase Coexistence · Clausius–Clapeyron Relation.

What is the difference between a tricritical point and an ordinary critical point?

An ordinary critical point terminates a first-order line where the two coexisting phases become identical, and it is reached by tuning one field (temperature) to a boundary. A tricritical point is where a line of continuous transitions meets a line of first-order transitions and the transition changes order; reaching it requires tuning two independent fields (codimension two). Formally, the ordinary case has a stabilizing quartic term ( $u>0$ ), while the tricritical case has $u=0$ and relies on the sextic term. The exponents differ: ordinary mean field gives $\beta=\tfrac12,\delta=3$ , tricritical gives $\beta_t=\tfrac14,\delta_t=5$ .

Why is the upper critical dimension 3 for tricriticality instead of 4?

Because the relevant nonlinearity is $\phi^6$ , not $\phi^4$ . Power counting assigns the sextic coupling scaling dimension $[w]=6-2d=2(3-d)$ , so it is marginal at $d=3$ (irrelevant above, relevant below). The general rule for a $\phi^{2n}$ interaction is $d_c=2n/(n-1)$ : $n=2$ (quartic) gives $d_c=4$ for ordinary criticality, $n=3$ (sextic) gives $d_c=3$ for tricriticality. This is why real three-dimensional tricritical points sit exactly at their marginal dimension and show mean-field exponents with only logarithmic corrections, while ordinary 3D critical points (below their $d_c=4$ ) show strongly non-classical exponents.

What plays the role of the second field in real experiments?

A non-ordering field $g$ that tunes the effective quartic coefficient $u$ through zero. In $^3$ He– $^4$ He mixtures it is the $^3$ He concentration; in metamagnets it is the uniform magnetic field (with the staggered magnetization the order parameter); in the Blume–Capel model it is the single-ion anisotropy $D$ that penalizes $S_i=\pm1$ relative to $S_i=0$ . In each case increasing this second field drives $u$ negative and converts the continuous transition into a first-order one, with the tricritical point at the crossover.

Do the tricritical exponents obey the same scaling relations as ordinary ones?

Yes — the scaling relations are structural and hold for both, only the numbers change. The tricritical set $\alpha_t=\tfrac12,\beta_t=\tfrac14,\gamma_t=1,\delta_t=5$ satisfies Rushbrooke $\alpha+2\beta+\gamma=2$ ( $\tfrac12+\tfrac12+1=2$ ) and Widom $\gamma=\beta(\delta-1)$ ( $\tfrac14\cdot4=1$ ), exactly as the ordinary set $\alpha=0,\beta=\tfrac12,\gamma=1,\delta=3$ does. What differs is the gap exponent ( $\Delta_t=\beta_t\delta_t=\tfrac54$ versus $\Delta=\tfrac32$ ) and the crossover exponent $\phi_t=\tfrac12$ that governs the size of the critical-to-tricritical crossover region — the latter has no analogue at an ordinary critical point.

Is a tricritical point the same as a critical endpoint?

No, and confusing them is a common error. At a tricritical point the same transition line smoothly changes from second to first order — one order parameter, one universality of behaviour along the line. A critical endpoint is different: it is where a line of critical points of one phase transition runs into a first-order coexistence surface involving a third phase and terminates there. The tricritical point is intrinsic to a single ordering transition; the critical endpoint is an intersection of distinct transitions. Their phase diagrams and Landau descriptions are different.

How does the crossover exponent φ_t enter the observations?

The crossover exponent $\phi_t=\tfrac12$ (mean field) sets the width of the region around the tricritical point where tricritical scaling is visible rather than ordinary critical scaling. Along the critical line but close to the tricritical point, the correlation length and other quantities cross over from ordinary critical behaviour (far away) to tricritical behaviour (close in) when the scaled second field $g/|t|^{\phi_t}$ becomes of order one. Experimentally this means the tricritical exponents are seen only in a narrow window; outside it one measures the ordinary critical exponents. Mapping this crossover is a standard way to locate the tricritical point precisely.