Hierarchical reference theory of fluids: Application to three-dimensional Ising model

The hierarchical reference theory (HRT) of fluids is applied to the three-dimensional Ising model on a simple cubic lattice with nearest-neighbor ferromagnetic interaction via the equivalence with the lattice-gas model. The hierarchy is truncated to the first equation and closed with an Ornstein-Zernike ansatz for the direct correlation function embodying both thermodynamic consistency and on-site repulsion between lattice particles. The resulting equations are integrated numerically above and below the critical temperature and the results are compared with those obtained by closed-form approximants. We show that HRT yields nontrivial critical exponents with the correct scaling regime and a value.of the critical temperature in very close agreement with the true one. At the same time it retains all the information about the short-range behavior of the system, and so gives a very accurate description also away from the critical point. Below the critical temperature as long as long-wavelength fluctuations are included in the system the van der Waals loop is suppressed and is replaced by a region where the compressibility is infinite, namely the coexistence region.