The formation of low-temperature superstructures in the two-dimensional Ising model with next-nearest neighbour interactions

For several decades the formation of different kinds of superstructures in solids has been a topical issue in condensed matter physics. The superstructures (or spatially modulated structures) may be of a different nature: magnetic patterns like spin-density waves, inhomogeneous charge distributions in charge-ordered compounds, dipolar and quadrupolar ordering in ferroelectrics or ferroelastics, regular lattice distortions and related orbital structures, stripe-like arrangements of dopants in alloys, etc. The phase diagrams of such compounds can be rather complicated involving a large number of phases with non-trivial types of ordering. Fortunately, all this wealth of seemingly unrelated phenomena can be often described by rather simple models with a due account taken of a competitive character of the most important interactions. In this thesis I will investigate the Ising model in 2-D with nearest and next-nearest neighbour interactions using several methods including exact diagonalisation of small clusters, transfer matrix technique and Monte Carlo simulation of large lattices. [Continues.]