We study the propagation of light through a semi-infinite medium with transverse disorder (that is, disorder in two directions only). We show that such a system exhibits strong two-dimensional localization by demonstrating that on propagation a beam expands until the transverse localization length is reached.

We present a detailed numerical study of the electronic properties of single-layer graphene with resonant (hydrogen) impurities and vacancies within a framework of noninteracting tight-binding model on a honeycomb lattice. The algorithms are based on the numerical solution of the time-dependent Schrödinger equation and applied to calculate the density of states, quasieigenstates, ac and dc conductivities of large samples containing millions of atoms. Our results give a consistent picture of evolution of electronic structure and transport properties of functionalized graphene in a broad range of concentration of impurities (from graphene to graphane), and show that the formation of impurity band is the main factor determining electrical and optical properties at intermediate impurity concentrations, together with a gap opening when approaching the graphane limit.

A theoretical analysis is given of the equation of motion method, due to Alben et al. [Phys. Rev. B 12, 4090 (1975)], to compute the eigenvalue distribution (density of states) of very large matrices. The salient feature of this method is that for matrices of the kind encountered in quantum physics the memory and CPU requirements of this method scale linearly with the dimension of the matrix. We derive a rigorous estimate of the statistical error, supporting earlier observations that the computational efficiency of this approach increases with the matrix size. We use this method and an imaginary-time version of it to compute the energy and specific heat of three different, exactly solvable, spin-1/2 models, and compare with the exact results to study the dependence of the statistical errors on sample and matrix size.

We study the properties of Suzuki's systematic approximations to the exponential operator $\exp (-\beta H)$ by calculating the thermodynamic functions of three simple quantum models. We demonstrate that the path-integral representation of the partition function obtained from these approximations can be simplified and made more accurate by constructing Hermitian versions of Suzuki's expressions.