Some spectral properties of the one-dimensional disordered Dirac equation

We study spectral properties of a one-dimensional Dirac equation with vario us disorder. We use replicas to calculate the exact density of state and ty pical localization length of a Dirac particle in several cases. We show tha t they can be calculated, in quite a simple fashion, in any type of disorde r obeying a Gaussian white noise distribution, In addition to cases involvi ng pure types of disorder, we study a mixed disorder case where the Dyson s ingularity is destroyed by the mixing. We also clarify the supersymmetric a lternative derivation, even though it proves less efficient than the replic a treatment for such thermodynamic quantities. We show that the smallest dy namical algebra in the Hamiltonian formalism is u(1, 1), preferably to u(n, n) in the replica derivation or u(1, 1\2) in the supersymmetric alternativ e. Finally, we discuss symmetries in the disorder fields and show that ther e exists a non-trivial mapping between the electric potential disorder and the magnetic (or mass) disorder. (C) 1999 Elsevier Science B.V.