The purpose of this thesis is the treatment of the antiferromagnetic Heisenberg model within the valence bond basis, allowing for the description of its low-energy physics. The manuscript is organized in two parts: In the first part we utilize the fidelity concept in order to detect quantum phase transitions. It is notably shown, that this quantity is accessible in a valence bond projector quantum Monte Carlo algorithm, making available the fidelity approach to large scale simulations. The second part is devoted to the generalization of an idea going back to Rokhsar and Kivelson, which aims to map a Heisenberg model onto a numerically less demanding quantum dimer model. Starting from a rigourous derivation, the method is then applied to the kagome lattice and allows to establish the existence of a tricritical point in the vicinity of the original model. The same technique is also used to treat the J1-J2-J3 Heisenberg model on the honeycomb lattice, showing the existence of a plaquette phase in a determined parameter regime.
The purpose of this thesis is the treatment of the antiferromagnetic Heisenberg model within the valence bond basis, allowing for the description of its low-energy physics. The manuscript is organized in two parts: In the first part we utilize the fidelity concept in order to detect quantum phase transitions. It is notably shown, that this quantity is accessible in a valence bond projector quantum Monte Carlo algorithm, making available the fidelity approach to large scale simulations. The second part is devoted to the generalization of an idea going back to Rokhsar and Kivelson, which aims to map a Heisenberg model onto a numerically less demanding quantum dimer model. Starting from a rigourous derivation, the method is then applied to the kagome lattice and allows to establish the existence of a tricritical point in the vicinity of the original model. The same technique is also used to treat the J1-J2-J3 Heisenberg model on the honeycomb lattice, showing the existence of a plaquette phase in a determined parameter regime.
In this thesis I present three publications about the use of valence bonds to gain information about quantum spin systems. Valence bonds are an essential ingredient of low energy states present in many compounds. The first part of this thesis is dedicated to two studies of the antiferromagnetic J1-J2 chain with S=1/2. We show how automated variational calculations based on valence bond states can be performed close to the Majumdar-Ghosh point (MG-point). At this point, the groundstate is a product state of dimers (valence bonds between nearest neighbours). In the dimerized region surrounding the MG-point, we find such variational computations to be reliable. The first publication is about the entanglement properties of an impurity attached to the chain. We show how to use the variational method to calculate the negativity, an entanglement measure between the impurity and a distant part of the chain. We find that increasing the impurity coupling and a minute explicit dimerization, suppress the long-ranged entanglement present in the system for small impurity coupling at the MG-point. The second publication is about a transition from commensurate to incommensurate behaviour and how its characteristics depend on the parity of the length of the chain. The variational technique is used in a parameter regime inaccessible to DMRG. We find that in odd chains, unlike in even chains, a very intricate and interesting pattern of level crossings can be observed. The publication of the second part is about novel worm algorithms for a popular quantum Monte Carlo method called valence bond quantum Monte Carlo (VBQMC). The algorithms are based on the notion of a worm moving through a decision tree. VBQMC is entirely formulated in terms of valence bonds. In this thesis, I explain how the approach of VBQMC can be translated to the Sz-basis. The algorithms explained in the publication can be applied to this Sz-method.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.
We present a comprehensive computational study of the phase diagram of the frustrated S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest (J2) and third-neighbor (J3) couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger J2, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.
We study the plaquette valence-bond solid phase of the spin-1/2J1-J2 antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four S=1/2 spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the J1-J2 model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region 0.34<J2/J1<0.59, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings J2=0.34J1 and J2=0.59J1. Interestingly, for J2<0.48J1, the excitation gap corresponds to a singlet-triplet excitation at the $\ensuremath{\Gamma}$ point while, for J2>0.48J1, it is related to a singlet-singlet excitation at the $\mathbf{X}=(\ensuremath{\pi}/2,0)$ point of the tetramerized Brillouin zone.
L'objet de cette thèse est le traitement du modèle de Heisenberg antiferromagnétique dans la base de liens de valence, qui permet d'en décrire la physique de basse énergie. Le manuscrit est organisé en deux parties : dans la première nous utilisons le concept de fidélité afin de détecter les transitions de phases quantiques. Nous démontrons notamment que cette quantité est accessible dans un algorithme de Monte Carlo quantique, formulé dans la base de liens de valence, permettant ainsi de calculer la fidélité sur des systèmes de grande taille. La deuxième partie vise à développer l'idée initiale de Rokhsar et Kivelson, qui a pour but de transformer un modèle de Heisenberg en un modèle de dimères quantiques, généralement moins complexe d'un point de vue numérique. Après une dérivation rigoureuse, cette technique est appliquée au réseau kagomé et permet d'établir l'existence d'un point tricritique au voisinage du modèle initial. La même méthode est ensuite utilisée afin de traiter le modèle J1-J2-J3 sur le réseau hexagonal et démontre l'existence d'une phase plaquette dans un domaine de paramètres déterminé.