Localized and Extended States in One-Dimensional Disordered System: Random-Mass Dirac Fermions

System of Dirac fermions with random-varying mass is studied in detail. We reformulate the system by transfer-matrix formalism. Eigenvalues and wave functions are obtained numerically for various configurations of random telegraphic mass m(x). Localized and extended states are identified. For quasi-periodic m(x), low-energy wave functions are also quasi-periodic and extended, though we are not imposing the periodic boundary condition on wave function. On increasing the randomness of the varying mass, states lose periodicity and most of them tend to localize. At the band center or the low-energy limit, there exist extended states which have more than one peak spatially separate with each other comparatively large distance. Numerical calculations of the density of states and ensemble averaged Green's functions are explicitly given. They are in good agreement with analytical calculations by using the supersymmetric methods and exact form of the zero-energy wave functions.