The liquid-gas critical point in 3.3 dimensions

Liquid-gas phase transitions are usually assumed to belong to the same universality class as Ising magnets. However, in the liquid-gas case, the free-energy expansion contains odd powers of the order parameter. We show in this work that, even though the odd terms are irrelevant near four dimensions, they become relevant at dimensionality $d_c=\frac{(10-5\mathrm{}\eta )}{3}\approx 3.33$. A new, stable, fixed point may appear. We approximately calculate, to order ${\epsilon }^{\prime }=d_c-d$, the difference between the critical exponents for this new fixed point and the Ising critical exponents. This difference turns out to be numerically small. We conclude that, even though in three dimensions the liquid-gas and Ising-magnet phase transition do not belong to the same universality class, the effect on the critical exponents is probably too small to be observed.