Perturbative Analysis for Random One-Dimensional Media around Hyperbolic Critical Energies

Quantum mechanical models on a one-dimensional lattice with random hopping terms that are restricted to be only nearest-neighbor are considered, e.g., such a random hopping model or the Su-Schrieffer-Heeger model. At a so-called critical energy level, the induced transfer matrices commute. If their (random) spectrum lies off the unit circle, such a critical energy is called hyperbolic. When in this case the Lyapunov exponent of the considered system vanishes, it is inversely proportional to the logarithm of a small deviation from this critical energy, and the integrated density of states is proportional to its square. The latter implies a characteristic divergence (Dyson spike) of the density of states at a hyperbolic critical energy, whereas a pseudogap is present if the Lyapunov exponent does not vanish. The application of techniques for random walks provide exact results and error bounds for the adopted perturbative approach.