Localized and extended states in one-dimensional disordered system: random-mass Dirac fermions

Abstract

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.

Random-Mass Dirac Fermions in an Imaginary Vector Potential: Delocalization Transition and Localization Length

“…For the vanishing imaginary vector potential, we can solve the Schrödinger equations in (4) under the periodic boundary condition with various multi-solitonantisoliton configurations of m(x) using the transfermatrix method(TMM). We can also obtain the energy spectrum and wave functions [10] using this method.…”

“…In the previous papers [10][11][12] we studied the effect of nonlocal correlation of the random mass on the extended states which exist near the band center. For numerical studies, we reformulate the system by transfer-matrix formalism, and obtained eigenvalues and wave functions for various configurations of random telegraphic mass [10]. We verified that the density of states obtained by the transfer-matrix methods is in good agreement with the analytical calculation by supersymmetric methods in Ref.…”

Quantum spin chains with nonlocally-correlated random exchange coupling and random-mass Dirac fermions

“…This localized massless mode corresponds to an unpaired spin in the dimer state of the spin chains which appears as a result of the sign change of m(n) in (5). In the randommass Dirac fermions, the random mass m(x) vanishes at various points x = x i (i = an integer) and roughly speaking low-energy modes are given by linear combinations of the localized modes in the vicinity of the kinks of m(x) [2] like i C i |i where |i is the localized state at x i and C i 's are coefficients. The long-range correlation of the random mass keeps distance between adjacent kinks |x i − x i+1 | longer and as a result it makes the LL of the low-energy states larger.…”

“…where σ are the Pauli spin matrices, m(x) is the "continuum limit" of m(n) (x = na with the lattice spacing a), ψ(x) = (ψ R (x), ψ L (x)) t and we have put J = 1 without loss of generality. The above Hamiltonian is nothing but the random-mass Dirac field in 1D which we studied in the previous papers [2,3]. The LL and DOS were calculated by the transfer-matrix methods and imaginary vector potential methods.…”

Asymptotic Solutions of a Discrete Schrödinger Equation Arising from a Dirac Equation with Random Mass

“…m(x)σ y can be considered as a random mass. This physical problem has some interesting properties in terms of the solution Ψ E (E) [8]. For instance, the solution is localized for E = 0 and/or m 0 = 0, i.e., the average wavefunction Ψ E (x) decays exponentially on a length scale ξ(E, m 0 ), and ξ(E, m 0 ) diverges at E = m 0 = 0.…”