A study of one dimensional correlated disordered systems using invariant measure method

The behavior of electronic states of one dimensional correlated disordered systems which are modelled by a tight binding Hamiltonian is studied analytically using the invariant measure method. The approach of Bovier is generalized to include the possibility of different site energies and nearest neighbor hopping integrals inside the correlated sites or the cluster. The process is further elaborated by applying to the symmetric random trimer model which contains in it many hitherto known models of this category. An alternative mathematical definition of the exceptional energy ($E_S$) from the invariant measure density along with physical arguments substantiating it is presented. Furthermore, the procedure for obtaining exceptional energies is outlined and applied to the symmetric random trimer model to derive conditions for obtaining doubly degenerate exceptional energies. The Lyapunov exponent ($\gamma (E)$) or the inverse localization length of states around the exceptional energy is found to vary as $\sim (E - E_S)^{2n}$ in the leading order. $n$ denotes the degeneracy of the exceptional energy. The density of states at the exceptional energies are calculated. We further propose that one dimensional correlated disordered systems can be mapped to a Lloyd model in which the width of the distribution of site energies is determined by the reflection coefficient of the cluster embedded in the lattice of the another constituent. The importance of our results is discussed.