Critical phenomena of fluids: Asymmetric Landau-Ginzburg-Wilson model

The critical behavior of simple single-component fluids near the liquid-vapor critical point (and of other systems in the same universality class) is examined via an asymmetric spin Hamiltonian in Landau-Ginzburg-Wilson form. The leading important odd terms to be added to an Ising-type system are a five-spin interaction $\int \varphi {}_{}{}^5{}_{}{}^{}\mathrm{}\left(x\right)\mathrm{}$ and a nonlocal cubic interaction $-\int \varphi {}_{}{}^2{}_{}{}^{}\nabla _{}^2{}_{}{}^{}\varphi$. These two operators can be combined into two different eigenoperators with distinct physical interpretations. One combination can be shown to be exactly equivalent to the mixing of field variables (e.g., chemical potential and temperature) in the temperaturelike variable of an Ising or Ising-type system. This mixing has been used phenomenologically, found in certain models, and justified on general geometric grounds. This is the first proof that all asymmetric models, assuming universality and barring accidents, will have such mixing; this implies the universal occurrence of a $t^{1-a}$ fluid diameter. The second combination is treated by renormalization-group methods and its contributions to the Helmholtz free energy, magnetic equation of state, and correlation length are calculated to $O\left({\epsilon }^2\right), \epsilon \equiv 4-d$. There are several novel features of this interaction which persist to all orders in perturbation theory. In addition to the expected renormalized $M^5$ contribution to the free energy, there are also terms linear in $h$ (the magnetic field) and $M$ which could be described as nonanalytic fluctuation-induced shifts in the order parameter and field. These might appear experimentally as apparent shifts in the background terms for $h$ and $M$.