Tricritical wing

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.