Exact and approximate differential renormalization-group generators. II. The equation of state

We apply an approximate form of the exact one-particle-irreducible renormalization-group generator to the calculation of the equation of state. Several approaches are explored. (i) Global nonlinear trajectories for the Hamiltonian parameters are exploited to give nonlinear crossover equations of state. Logarithms of the reduced temperature $t\equiv \frac{(T-T_c)}{T_c}$ are automatically exponentiated to give power-law behavior. (ii) The form and asymptotic properties of the equation of state are described for those systems whose complete nonlinear trajectories cannot be explicitly obtained. (iii) Nonlinear trajectories for the irreducible Green's functions are solved to give a fully exponentiated equation of state containing no logarithmic terms. (iv) Simple critical (noncrossover) equations of state are obtained by iteration of the generator with linear and quasilinear trajectories. (v) Operator nonlinear equations are solved to give the crossover equation of state for an arbitrary order $\theta$ critical system, both in an $\epsilon$ expansion and at the borderline dimension. (vi) Combination of the Green's functions are formed into Green's eigenfunctions or operators and their nonlinear trajectories used to calculate fully exponentiated equations of state for order $\theta$ Ising systems. Systems considered include the usual Wilson-Fisher Hamiltonian [methods (i)-(iv)], the Sak-model compressible ferromagnet [(i)-(ii)], the $\mathrm{nm}$-hypercubical model (of which the dilute quenched random ferromagnet forms the $n\to 0$ limit) [(ii)], and the isotropic $n$-component order $\theta$ ferromagnet [(iv)-(vi)]. The tricritical ($\theta =3\mathrm{}$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.