A hybrid lattice-statistical model of doubly decorated two-dimensional lattices, which have localized Ising spins at its nodal sites and itinerant electrons delocalized over decorating sites, is exactly solved with the help of a generalized decoration-iteration transformation. Under the assumption of a quarter filling of each couple of the decorating sites, the ground state constitutes either spontaneously long-range ordered ferromagnetic or ferrimagnetic phase in dependence on whether the ferromagnetic or antiferromagnetic interaction between the localized Ising spins and itinerant electrons is considered. The critical temperature of the spontaneously long-range ordered phases monotonically increases upon strengthening the ratio between the kinetic term and the Ising-type exchange interaction.
In the paper the Pair Approximation (PA) method for studies of the site-diluted spin-1/2 systems of arbitrary dimensionality with the long-range ferromagnetic interactions is adopted. The method allows to take into account arbitrary anisotropy of the interactions in the spin space, so it is not limited to purely Ising couplings. Within this approach, the Gibbs free energy is obtained, which allows to derive all the further interesting thermodynamic properties. In particular, we obtain an equation for the critical temperature of the second-order phase transitions for the model in question. In the study we focus our attention on the systems with ferromagnetic interactions decaying with the distance according to the power law
J(r)∝r−n. We discuss the dependence of the critical temperature on the concentration of magnetic component and the index
n for selected one-, two- and three-dimensional lattices.We confirm the absence of the critical concentration for a diluted magnet with infinite interaction range. In the regime of the low concentrations of magnetic component, we find a non-linear increase of the critical temperature with the concentration in the form of
Tc∝pn/d, depending on the system dimensionality
d and the index
n, whereas
n>d.
We study the effect of long-range connections on the infinite-randomness fixed point associated with the quantum phase transitions in a transverse Ising model (TIM). The TIM resides on a long-range connected lattice where any two sites at a distance
r are connected with a nonrandom ferromagnetic bond with a probability that falls algebraically with the distance between the sites as $1∕{r}^{d+\ensuremath{\sigma}}$. The interplay of the fluctuations due to dilutions together with the quantum fluctuations due to the transverse field leads to an interesting critical behavior. The exponents at the critical fixed point (which is an infinite randomness fixed point) are related to the classical ``long-range'' percolation exponents. The most interesting observation is that the gap exponent $\ensuremath{\psi}$ is exactly obtained for all values of $\ensuremath{\sigma}$ and
d. Exponents depend on the range parameter $\ensuremath{\sigma}$ and show a crossover to short-range values when $\ensuremath{\sigma}\ensuremath{\ge}2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}$ where ${\ensuremath{\eta}}_{\mathrm{SR}}$ is the anomalous dimension for the conventional percolation problem. Long-range connections are also found to tune the strength of the Griffiths phase.