The recursion relations obtained by applying the renormalization group technique to a finite one-dimensional Ising chain with nearest-neighbor interaction and constant external magnetic field are decoupled and solved in closed form.
We construct a renormalization-group theory for a one-dimensional Ising model with longranged interactions and random fields with short-ranged correlations. We find recursion relations near the zero-temperature, finite randomness fixed point, valid for $\ensuremath{\sigma}$ near \textonehalf{}, where the interaction $\frac{J}{(r)}=\frac{J}{{r}^{1+\ensuremath{\sigma}}}$. We also show that the probability distribution for the random fields becomes strongly non-Gaussian under the scaling procedure.
Sharp two- and three-dimensional phase transitional magnetization curves are obtained by an iterative renormalization-group coupling of Ising chains, which are solved exactly. The chains by themselves do not have a phase transition or nonzero magnetization, but the method reflects crossover from temperaturelike to fieldlike renormalization-group flows as the mechanism for the higher-dimensional phase transitions. The magnetization of each chain acts, via the interaction constant, as a magnetic field on its neighboring chains, thus entering its renormalization-group calculation. The method is highly flexible for wide application.
The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.