Real-space simulation of two-dimensional interacting quantum condensed matter systems

The proliferation of quantum fluctuations and long-range entanglement presents an outstanding challenge for the numerical simulation of quantum condensed matter systems with exotic ground states. In this thesis, I tackle two classes of two-dimensional interacting models on the honeycomb lattice: multi-orbital Hubbard models on zig zag transition metal dichalcogenide nanoribbons and generalised Kitaev models on periodic clusters. In the first part of the thesis, I discuss novel results obtained in a comparative study of mean field theory (MFT) and determinant quantum Monte Carlo (DetQMC). MFT reveals the influence of the edge filling on the ground state of the ribbons. The unbiased, numerically exact DetQMC confirms the stability of one of the possible ground states, albeit with quantitative differences, such as the critical Hubbard interaction for the onset of magnetic order. Unfortunately, DetQMC is severely plagued by the sign problem for this model. The variance of its estimators grows exponentially as most of the relevant edge fillings are reached and simulations are rendered unfeasibly expensive from the computational standpoint. Motivated by the difficulties posed by the sign problem, I carry out a survey of general purpose numerical methods. The second part of the thesis addresses quantum spin liquids — which have attracted increasing attention — presenting a toolset of Chebyshev spectral methods developed here, namely: the finite temperature Chebyshev polynomial and the hybrid Lanczos-Chebyshev methods. The first one enables studies of temperature dependence for quantities of experimental interest, such as the specific heat, with a two-fold speed-up with respect to state-of-the-art methods. The second one gives access to spectral functions efficiently and with unparalleled flexibility. I use it to obtain novel results for the spin susceptibility of the Kitaev-Ising model, unravelling dynamical signatures of a liquid–to–liquid transition. Finally, I briefly discuss the integration of the novel Chebyshev toolset with existing open-source software.