Exact solutions to many-body interacting systems of both bosonic and fermionic particles confined to harmonic potential in an arbitrary number of dimensions are given. Energy levels and their degeneracies for trapped identical particles interacting via harmonic forces are calculated. This specific form of the interaction allows for analytical solutions. The mutual interaction, attractive or repulsive, modifies significantly the properties of the considered system. For a large number of particles the interaction essentially results in a frequency shift. Statistical properties (e.g., microcanonical and grand canonical partition functions) as well as some illustrative, physically relevant examples are discussed. Our results give an unusual opportunity for further studies of interacting systems in the framework of the exactly soluble model.

Bunching and meandering instability of steps at the 4H-SiC(0001) surface is studied by the kinetic Monte Carlo simulation method. Change in the character of step instability is analyzed for different rates of particle jumps towards step. In the experiment effective value of jump rates can be controlled by impurities or other growth conditions. An anisotropy of jump barriers at the step influences the character of surface structure formed in the process of crystal growth. Depending on the growth parameters different surface patterns are found. We show phase diagrams of surface patterns as a function of temperature and crystal growth rate for two different choices of step kinetics anisotropy. Jump rates which effectively model high inverse Schwoebel barrier (ISB) at steps lead either to regular, four-multistep or bunched structure. For weak anisotropy at higher temperatures or for lower crystal growth rates meanders and mounds are formed, but on coming towards lower temperatures and higher rates, we observe bunch and meander coexistence. These results show that interplay between simple dynamical mechanisms induced by the asymmetry of the step kinetics and step movement assisted by the step edge diffusion are responsible for different types of surface morphology.

Growth of gallium nitride on GaN(0001) surface is modeled by Monte Carlo method. Simulated growth is conducted in N-rich conditions, hence it is controlled by Ga atoms surface diffusion. It is shown that dominating four-body interactions of Ga atoms can cause step flow anisotropy. Kinetic Monte Carlo simulations show that parallel steps with periodic boundary conditions form double terrace structures, whereas initially $V$ -shaped parallel step train initially bends and then every second step moves forward, building regular, stationary ordering as observed during metal organic vapor phase epitaxy or hydride vapor phase epitaxy growth of GaN layers. These two phenomena recover surface meandered pair step pattern observed, since 1953, on many semiconductor surfaces, such as SiC, Si, or GaN. Change in terrace width or step orientation particle diffusion jump barriers leads either to step meandering or surface roughening. Additionally it is shown that step behavior changes with the Schwoebel barrier height. Furthermore, simulations under conditions corresponding to very high external particle flux result in triangular islands grown at the terraces. All structures, emerging in the simulations, have their corresponding cases in the experimental results.

A realization of the numerous phases predicted in systems with long-range repulsive interactions was recently found in $\mathrm{Pb}/\mathrm{Si}(111)$. Surprisingly, these numerous phases can be grown at low temperatures $\sim 40\mathrm{K}$ over macroscopic distances. This unusual observation can be explained from theoretical calculations of the collective diffusion coefficient $D_c$ in systems with long-range repulsive interactions. Instead of a gradual dependence of $D_c$ on coverage, it was found that $D_c$ has sharp maxima at low temperatures for every stable phase (i.e., for every rational value of the coverage $\theta =p/q$) in agreement with the experiment.