Locally ordered regions in the phase transition in the systems with a finite-range correlated quenched disorder

The locally ordered regions (LOR) in the phase transition in disordered systems are studied. There are two parts in this paper. One part is to report our numerical results on the one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature in the presence of an ordering field. The disordered system is modelled as a lattice, on which each cell has a local reduced temperature. The random part of the local reduced temperature is distributed in the Gaussian form. The one-dimensional saddle point equation is solved numerically. The average, the fluctuation and the correlation length of the solution are calculated. The scaling relations for these quantities with the temperature, the ordering field and the disorder strength are derived. The numerical data are fitted with the scaling relations well. Another part is to discuss qualitatively the phase diagram of the finite-range correlated disordered systems. There are two proposed classes for the phase transition in connection with the LOR. One class is described by the percolative scenario, in which the phase transition is inhomogeneous. In the percolative scenario the percolation of the LOR dominates the phase transition. In another class, the phase transition is homogeneous, and can be described by the renormalization group (RG) with replica symmetry breaking (RSB). In the RG with RSB, there is nothing to do with the percolation of LOR. We shall show that these two theories, which seem contradictory, may describe two parts of the whole phase diagram. Whether the phase transition is homogeneous or inhomogeneous depends on the interaction between the LOR. If the interaction between the LOR is strong enough, the phase transition is percolative and inhomogeneous. If the interaction between the LOR is weak, the phase transition is homogeneous. The interaction between the LOR is discussed with the numerical solution on the saddle point equation.