text: A first-order transition will arise if the quartic term in  is negative. To ensure that the free energy remains positive at large , one must carry the free-energy expansion to sixth-order,
  where , and  is some temperature at which  changes sign. We denote this  temperature by  and not , since it will emerge below that it is not the temperature of the first-order transition, and since there is no critical point, the notion of a "critical temperature" is misleading to begin with.  and  are positive coefficients.
  We analyze this free energy functional as follows: (i) For , the  and  terms are concave upward for all , while the  term is concave downward. Thus for sufficiently high temperatures  is concave upward for all , and the equilibrium solution is . (ii) For , both the  and  terms are negative, so  is a local maximum, and the minimum of  is at some non-zero value , with
  . (iii) For  just above ,  turns into a local minimum, but the minimum at  continues to be the global minimum since it has a lower free energy. It follows that as the temperature is raised above , the global minimum cannot continuously evolve from
  to 0. Rather, at some intermediate temperature , the minima at  and  must become degenerate. For , the global minimum will jump discontinuously from  to 0.