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The mean-field approximation suppresses correlated order-parameter fluctuations. Near a continuous phase transition, the correlation length diverges and long-wavelength fluctuations become increasingly important. The Ginzburg criterion therefore fails in sufficiently low dimension, and the interacting renormalization-group fixed point changes the mean-field exponents. For the tricritical $m^6$ theory the upper critical dimension is three: below $d=3$ the exponents are generally non-mean-field, while at $d=3$ one expects logarithmic corrections to mean-field scaling.

Write $a={\alpha }_2(T)$ and $u={\alpha }_{2n}>0$. At high temperature $a>0$, so $f(m)=a{m}^2+u{m}^{2n}$ has one minimum at $m=0$. At low temperature $a<0$, the origin is a local maximum and two symmetry-related minima appear. The stationary equation isso the equilibrium magnetization isandBecause $a(T)\sim T-T_c$, the two nonzero minima approach zero continuously as $T\uparrow T_c$. The order parameter is continuous while its response becomes singular, so this is a continuous phase transition.

The stationary points obeyNonzero stationary points exist whenso they first appear at the ordered-phase spinodal point $a_2=9a_3^2/(32a_4)$. The disordered state $m=0$ is locally stable for $a_2>0$ and loses that stability at $a_2=0$.

The actual phase boundary is found by requiring a nonzero stationary point $m_{\ast }$ to have the same free energy as $m=0$. Solving $f^{\prime }(m_{\ast })=0$ and $f(m_{\ast })=0$ givesThe disordered state is the global minimum for $a_2>a_3^2/(4a_4)$, the ordered state is the global minimum for $a_2<a_3^2/(4a_4)$, and they coexist at equality.

Since $a_3\ne 0$, the order parameter jumps from zero to $-a_3/(2a_4)$ at coexistence. Hence this model has no continuous phase transition as the phases exchange stability, but it does have a first-order phase transition at the displayed positive value of $a_2$.