Some spectral properties of the one-dimensional disordered Dirac equation

We study spectral properties of a one-dimensional Dirac equation with various disorder. We use replicas to calculate the exact density of state and typical localization length of a Dirac particle in several cases. We show that they can be calculated, in quite a simple fashion, in any type of disorder obeying a Gaussian white noise distribution. In addition to cases involving pure types of disorder, we study a mixed disorder case where the Dyson singularity is destroyed by the mixing. We also clarify the supersymmetric alternative derivation, even though it proves less efficient than the replica treatment for such thermodynamic quantities. We show that the smallest dynamical 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 alternative. Finally, we discuss symmetries in the disorder fields and show that there exists a non-trivial mapping between the electric potential disorder and the magnetic (or mass) disorder.