the qualitative results are almost the same as the above two cases. But if $v_4$ become negative, the solution structure near the transition point where $\sigma$ turns to zero becomes different. As an example, we take $v_4=-2, v_6=100, \mu=0.5\rm{GeV}$ and show the results in Fig.\[simplev4+v6\]. In this situation, we could see that a triple-valued region of $\sigma$ appears in a narrow temperature region $0.222\rm{GeV}<T<0.223\rm{GeV}$, which indicates a first order phase transition. Though the transition order depends only on the sign of $v_4$ and is independent on $\mu$, we see the possibility of constructing a model to describe different transition orders in different region of $\mu$ by changing the sign of the sub-leading power term $v_4$. If $v_4$ effectively changes its value from positive at small $\mu$ to negative at large $\mu$, one might expect a change of transition order. It is easy to see that if one adds a term like $$\begin{aligned}
\alpha F^2(X^+X)^2\end{aligned}$$ to the action, effectively, the coefficient of $\chi^4$ term becomes $$\begin{aligned}
-\lambda_4A_t^{\prime2}\frac{z^4}{e^{4A_s}}+ v_4.\end{aligned}$$ Here, we have inserted $A_t=A_t(z), A_z=0$ and $A_i=0$ for spatial components of the gauge potential. When $\mu=0$, we take $v_4>0$, the transition is second order; while when $\mu$ is sufficient large, $-\lambda_4A_t^{\prime2}\frac{z^4}{e^{4A_s}}+ v_4$ 