Tricritical point

A tricritical point is where a line of continuous transitions meets a line of first-order transitions; in a ${\mathbb{Z}}_2$ Landau expansion both quadratic and quartic coefficients vanish and a positive sextic term stabilizes the free energy.

For the sextic Landau free energy $V=r{M}^2/2+v{M}^6/6-hM$, $v>0$, with $r$ linear in reduced temperature, the equation of state is $h=rM+v{M}^5$. Its ordered zero-field minimum satisfies $M^4=-r/v$, the inverse magnetic susceptibility is $4|r|$ below the transition and $r$ above, and the critical isotherm satisfies $h=v{M}^5$. The minimized singular free-energy density is $-|r{|}^{3/2}/(3\sqrt{v})$ below and zero above. These facts give the displayed mean-field critical exponents, obeying both the Rushbrooke scaling relation and Widom scaling relation. Both the quadratic and quartic coefficients must be tuned; a generic path with a positive quartic coefficient instead has ordinary critical behavior. Fluctuations change the result below the tricritical upper critical dimension three, and marginal corrections can add logarithms at three.

For $F=F_{\mathrm{reg}}+a{m}^2/2+c{m}^6/6$, $c>0$, $a=a_0(T-T_c)$, the ordered global minimum gives $F-F_{\mathrm{reg}}=-(-a{)}^{3/2}/(3\sqrt{c})$ and $C_{\mathrm{sing},-}=T{a}_0^2/(4\sqrt{c}\sqrt{-a})$. On the disordered side the minimum is exactly $m=0$, so the polynomial's heat capacity is zero. The tricritical scaling label $\alpha =1/2$ can be used with zero upper-side amplitude, but no nonzero upper-side power law follows from this saddle alone. A smooth background such as $F_{\mathrm{reg}}=-C_{0}T\log (T/T_{\ast })$ gives a finite upper-side heat capacity $C_0$ without changing the phase diagram.

For $F=F_0+b(T-T_c)M^2+d^{\prime }{M}^6$ with $b,d^{\prime }>0$, the ordered minimum has $M^4=b(T_c-T)/(3d^{\prime })$. Its singular free energy is $-2b^{3/2}(T_c-T{)}^{3/2}/\sqrt{27d^{\prime }}$. Twice differentiating in temperature gives the displayed ordered-side heat capacity singularity, with mean-field critical exponent $\alpha =1/2$. This path keeps the quartic coefficient tuned to zero; a generic path with a positive quartic term instead crosses over to the ordinary quartic result.

For an even sextic Landau free energy perturbed by a conjugate field, a tricritical wing is a phase coexistence surface at nonzero field between ordered minima of different magnitudes. Two symmetry-related wings emerge from the three-phase line at zero field and terminate at ordinary critical edges. They meet the zero-field ordered coexistence sheet at the tricritical point.

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.

Take two nonnegative coexisting minima $a\le b$, put $s=a+b$ and $p=ab$, and fix $v>0$. For the sextic Landau free energy with quadratic, quartic, sextic and linear terms, equal stationary minimum values occur atAt these coefficients,Thus both are global minima. The case $a=0$ gives the zero-field three-phase line, including the negative minimum; the limit $a=b$ gives the tricritical wing critical edge. Reflecting $M,h$ gives the negative-field wing.

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.

For the stable sextic Landau free energy $f=r{m}^2/2+q{m}^4/4+s{m}^6/6-hm$, $s>0$, rescaling $m=(|r|/s{)}^{1/4}\psi$ gives the minimized order-parameter contribution $f_{\mathrm{eq}}=|r{|}^{3/2}{s}^{-1/2}{\Phi }_{\sigma }(X,Y)$, where $\sigma =\mathrm{sgn}r$, $X=q/(|r|s{)}^{1/2}$ and $Y=h{s}^{1/4}/|r{|}^{5/4}$. The two functions are ${\Phi }_{\sigma }={\min }_{\psi }(\sigma {\psi }^2/2+X{\psi }^4/4+{\psi }^6/6-Y\psi )$. In particular ${\Phi }_{-}(0,0)=-1/3$, but ${\Phi }_{+}(0,0)=0$. These separate branches and removal of smooth background terms are necessary when using the homogeneous form to infer mean-field critical exponents.

On the ordered branch of tricritical crossover scaling, at zero field and $X\to +\infty$, the minimizing square is $a={\psi }^2=(\sqrt{X^2+4}-X)/2\sim 1/X$. Its stationarity relation $Xa+a^2=1$ gives ${\Phi }_{-}(X,0)=-a/4-a^3/12\sim -1/(4X)$. Thus the minimized free-energy density becomes $-r^2/(4q)$ for fixed $q>0$ as $r\uparrow 0$, recovering the ordinary quartic continuous phase transition and its heat-capacity critical exponent $\alpha =0$. Replacing the scaling function by a nonzero constant in this limit would give an incorrect exponent.

In three dimensions, after tuning the quadratic and quartic relevant operators, the positive sextic coupling has a renormalization-group beta function $d\lambda /ds=-C{\lambda }^2+\cdots$ with $C>0$ for increasing length scale $s$. The Wick contractions connecting two sextic vertices with three propagators give the primitive logarithmic correction. The interaction is marginally irrelevant.