[en] Magnetic frustration is a phenomenon arising in spin systems when spin interactions cannot all be satisfied at the same time. A typical example of geometric frustration is a triangle with Ising-spins at its vertices and antiferromagnetic interaction. While we can easily anti-align two neighbouring spins, it is not possible for the third one to simultaneously anti-align with both of them. Another flavour of magnetic frustration is the so called exchange frustration, where different spin components interact in an Ising fashion on different bonds. Moreover, frustrated spin systems give rise to exotic states of matter, such as spin liquids, spin ices and nematic phases. As frustrated systems are rarely analytically solvable, numerical techniques are of the utmost importance in this framework. This dissertation is concerned with a specific class of models, namely one- and quasi-one-dimensional spin systems and studies their properties by making use of the density matrix renormalisation group technique. This method has been shown to be extremely powerful and reliable to study chain and ladder models. We consider examples of both geometric and exchange frustration. For the former, we take into consideration one of the prototypical examples of geometric frustration in one dimension: the J${}_1$-J${}_2$ model with ferromagnetic nearest-neighbour interaction J${}_1$< 0 and antiferromagnetic next-nearest-neighbour interaction J${}_2$ > 0. Our results show the existence of a Haldane gap supported by a special AKLT-like valence bond solid state in a specific region of the coupling ratio. Furthermore, we consider the effect of dimerisation of the first-neighbour coupling. This dimerisation affects the critical point and the ground state underlying the spin gap. These models are of interest in the context of cuprate chain materials such as LiVCuO${}_4$, LiSbCuO${}_4$ and PbCuSO${}_4$(OH)${}_2$. Concerning exchange frustration, we consider the celebrated Kitaev-Heisenberg model: it is an extension of the exactly solvable Kitaev model with an additional Heisenberg interaction. The Kitaev-Heisenberg model is currently the minimal model for candidate Kitaev materials. The extended model is not analytically solvable and numerics are needed to study the properties of the system. While both the original Kitaev and the Kitaev-Heisenberg models live on a honeycomb lattice, we here perform systematic studies of the Kitaev-Heisenberg chain and of the two-legged ladder. While the chain cannot support a Kitaev spin liquid state, it shows nevertheless a rich phase diagram despite being a one-dimensional system. The long-range ordered states of the honeycomb can be understood in terms of coupled chains within the Kitaev-Heisenberg model. Following this reasoning, we turn our attention to the Kitaev-Heisenberg model on a two-legged ladder. Remarkably, the phase diagram of the ladder is extremely similar to that of the honeycomb model and the differences can be explained in terms of the different dimensionalities. In particular, the ladder exhibits a topologically non-trivial phase with no long-range order, i.e., a spin liquid. Finally, we investigate the low-lying excitations of the Kitaev-Heisenberg model for both the chain and the ladder geometry.

[en] We study the one-dimensional Kitaev-Heisenberg model as a possible realization of magnetic degrees of freedom of the K-intercalated honeycomb-lattice ruthenium trichloride α-RuCl₃, denoted as K_0.5RuCl_m. First, we discuss the possible charge ordering patterns in K_0.5RuCl_m, where half of the $j=\frac{1}{2}$ spins are replaced by nonmagnetic ions in the honeycomb layer. The phase diagram is obtained as a function of nearest-, second-, and third-neighbor repulsions in the atomic limit. We find a strong competition between possible three charge ordering patterns when the repulsions are comparable. We further calculate the spin-spin correlation functions for the 1D Kitaev-Heisenberg model, which is one of the ordering patterns, and consider the effect of the Kitaev term by comparing the results to those for the antiferromagnetic SU(2) Heisenberg chain. (paper)

[en] We study the one-dimensional Kitaev-Heisenberg model as a possible realization of magnetic degrees of freedom of the K-intercalated honeycomb-lattice ruthenium trichloride $\alpha$-${\mathrm{RuCl}}_3$, denoted as ${\mathrm{K}}_{0.5}$${\mathrm{RuCl}}_m$. First, we discuss the possible charge ordering pattern in ${\mathrm{K}}_{0.5}$${\mathrm{RuCl}}_m$, where half of the $j=\frac{1}{2}$ spins are replaced by nonmagnetic ions in the honeycomb layer. Next, we investigate the low-energy excitations of the 1D Kitaev-Heisenberg model by calculating the dynamical spin structure factor using the Lanczos exact-diagonalization method. In the vicinity of Kitaev limit, there exist two well-separated dispersions. The bandwidth of each dispersion depends on the Heisenberg and Kitaev terms. This result may be relevant to the low-lying magnetic excitations of ${\mathrm{K}}_{0.5}$${\mathrm{RuCl}}_m$.