Dissertations / Theses on the topic 'Fermionic lattice models'

In una formulazione rigorosa della teoria quantistica, la definizione della varietà Riemanniana spaziale su cui il sistema è vincolato gioca un ruolo fondamentale. La presenza di un bordo sottolinea l'aspetto quantistico del sistema: l'imposizione di condizioni al contorno determina la discretizzazione degli autovalori del Laplaciano, come accade con condizioni note quali quelle periodiche, di Neumann o di Dirichlet. Tuttavia, non sono le uniche possibili. Qualsiasi condizione al bordo che garantisca l'autoaggiunzione dell' operatore Hamiltoniano è ammissibile. Tutte le possibili boundary conditions possono essere catalogate a partire dalla richiesta di conservazione del flusso al bordo della varietà. Alcune possibili condizioni al contorno, permettono l'esistenza di stati legati al bordo, cioè autostati dell' Hamiltoniana con autovalori negativi, detti edge states. Lo scopo di questa tesi è quello di investigare gli effetti di bordo in sistemi unidimensionali implementati su un reticolo discreto, nella prospettiva di capire come simulare proprietà di edge in un reticolo ottico. Il primo caso considerato è un sistema di elettroni liberi. La presenza di edge states è completamente determinata dai parametri di bordo del Laplaciano discreto. Al massimo due edge states emergono, e possono essere legati all' estremità destra o sinistra della catena a seconda delle condizioni al contorno. Anche il modo in cui decadono dal bordo al bulk e completamente determinato dalla scelta delle condizioni. Ammettendo un' interazione quadratica tra siti primi vicini, un secondo tipo di stati emerge in relazione sia alle condizioni al contorno che ai parametri del bulk. Questi stati sono chiamati zero modes, in quanto esiste la possibilità che siano degeneri con lo stato fondamentale. Per implementare le più generali condizioni al contorno, specialmente nel caso interagente, è necessario utilizzare un metodo generale per la diagonalizzazione, che estende la tecnica di Lieb-Shultz-Mattis per Hamiltoniane quadratiche a matrici complesse.

Les systèmes fortement corrélés présentent souvent des diagrammes de phase riches avec différentes phases ordonnées impliquant des degrés de liberté de spin, de charge, d'appariement ou d'orbitale. La description théorique de la compétition entre les différentes instabilités dans les systèmes fortement corrélés, qui donne lieu à cette phénoménologie, reste l'un des Saint-Graal de la théorie moderne de la matière condensée. Elle pose un énorme défi de complexité à la fois conceptuelle et numérique, et l'interaction des fluctuations électroniques concurrentes constitue donc un obstacle à la compréhension des diagrammes de phase complexes d'une large gamme de matériaux quantiques corrélés. Cela motive la recherche de méthodes simplifiées pour étudier l'interaction des fluctuations collectives.Nous présentons une extension multicanal de l'approche du champ fluctuant récemment développée pour les fluctuations collectives concurrentes dans les systèmes électroniques corrélés. La méthode est basée sur une optimisation variationnelle d'une action d'essai qui contient explicitement les paramètres d'ordre des principaux canaux de fluctuation. Elle donne un accès direct à l'énergie libre du système, facilitant la distinction entre les phases stables et métastables du système. Nous appliquons notre approche au modèle de Hubbard étendu, un modèle de fermions sur réseau paradigmatique, qui occupe une place de choix dans la théorie de la matière condensée en raison de la pertinence potentielle de ses versions répulsives et attractives pour les matériaux électroniques et les systèmes artificiels. En utilisant notre technique pour étudier le régime de couplage faible à intermédiaire de l'interaction répulsive, nous constatons qu'elle capture la compétition entre les fluctuations d'onde de densité de charge et des fluctuations antiferromagnétiques en accord qualitatif avec des méthodes numériquement plus coûteuses. En outre, cette méthode permet d'accéder aux propriétés des états excités et aux effets de corrélation à plusieurs corps, directement sur l'axe des fréquences réelles sans utiliser de techniques de continuation analytique numériques. L'approche du champ fluctuant multicanal offre donc une voie prometteuse pour un traitement numériquement peu coûteux de l'interaction entre les fluctuations collectives dans les systèmes de petite et grande taille.En utilisant l'approche introduite du champ fluctuant multicanal, nous explorons le diagramme de phase du modèle de Hubbard étendu dans les régimes répulsif et attractif, en abordant l'interaction des fluctuations dans les canaux antiferromagnétiques, de l'onde de densité de charge, de l'onde s supraconductrice et de la séparation de phases. Bien que ce modèle ait été étudié de manière intensive depuis des décennies, notre nouvelle approche nous permet d'identifier une nouvelle phase caractérisée par la coexistence de fluctuations collectives de l'onde s supraconductrice et de la séparation de phases. Ces résultats sont en accord avec les observations précédentes de phases supraconductrices et de séparation de phases dans les systèmes électroniques, notamment dans les supraconducteurs à haute température critique. En outre, la méthode des champs fluctuants multicanaux permet de mettre en évidence la quintessence du modèle de Hubbard étendu grâce à la grande variété de types de compétitions qui émergent des différentes instabilités. La nature générale de la théorie proposée, qui permet d'incorporer une grande variété de modes collectifs, en fait un outil prometteur pour l'étude de l'interaction des fluctuations collectives dans les systèmes fermioniques fortement corrélés
Strongly correlated systems often display rich phase diagrams exhibiting different ordered phases involving spin, charge, pairing, or orbital degrees of freedom. The theoretical description of the competition between different instabilities in strongly correlated systems giving rise to this phenomenology, remains one of the holy grails of modern condensed matter theory. It poses a tremendous challenge of both conceptual and computational complexity, and thus the interplay of competing electronic fluctuations constitutes a roadblock to the understanding of the complex phase diagrams of a wide range of correlated quantum materials. This motivates the search for constructing simplified methods to study interplaying collective fluctuations.We introduce a multichannel extension of the recently developed fluctuating field approach to competing collective fluctuations in correlated electron systems. The method is based on a variational optimization of a trial action that explicitly contains the order parameters of the leading fluctuation channels. It gives direct access to the free energy of the system, facilitating the distinction between stable and metastable phases of the system.We apply our approach to the extended Hubbard model, a paradigmatic fermionic lattice model, occupying a prime place in condensed matter theory due to the potential relevance of its repulsive and attractive versions for both electronic materials and artificial systems.Utilising the technique to study the weak to intermediate coupling regime of the repulsive interaction, we find it to capture the interplay of competing charge density wave and antiferromagnetic fluctuations with qualitative agreement with more computationally expensive methods. In addition, the method allows access to excited-state properties, through the one-particle excitation spectrum, and many-body correlation effects, through the self-energy, directly on the real-frequency axis without using numerical analytic continuation techniques. The multichannel fluctuating field approach thus offers a promising route for a numerically low-cost treatment of the interplay between collective fluctuations in small to large systems.Using the introduced multichannel fluctuating field approach, we explore the phase diagram of the extended Hubbard model in both repulsive and attractive regimes, addressing the interplay of fluctuations in the antiferromagnetic, charge density wave, s-wave superconducting, and phase separation channels. Despite the fact that this model has been intensively studied for decades, our novel approach allows us to identify a novel phase that is characterised by the coexistence of collective s-wave superconducting and phase separation fluctuations. These findings resonate with previous observations of interplaying phase separation and superconducting phases in electronic systems, most importantly in high-temperature superconductors. In addition, the multichannel fluctuating field method allows to display the quintessential nature of the extended Hubbard model through the large variety of types of competitions which emerges from the interplaying instabilities. The general nature of the proposed theory, allowing to incorporate a variety of collective modes, makes it a promising tool for studying the interplay of collective fluctuations in strongly correlated fermionic systems

The first part deals with lattice approach to field theories. The fermion doubling problems are described. This doubling can be removed if a dual lattice is introduced, as first pointed out by Stacey. His method is developed and in the process a formalism for the construction of a covariant difference lattice operator and thus of a gauge invariant action, is exhibited. It is shown how this formalism relates to the work of Wilson. Problems of gauge invariance can be traced back to the absence of the Leibnitz rule on the lattice. To circumvent this failure the usual notion of the product is replaced by a convolution. The solutions display a complementarity : the more localised the product the more extended is the approximation to the derivative and vice-versa. It is found that the form of the difference operator in the continuous limit dictates the formulation of the full two-dimensional supersymmetric algebra. The construction of the fields necessary to form the Wess-Zumino model follows from the requirement of anticommutativity of the supersymmetric charges. In the second part, the Skyrme model is reviewed and Bogomolnyi conditions are defined and discussed. It appears that while the Skyrme model has many satisfactory features, it fails to describe the interactions between nucleons correctly. These problems are brought out and the available solutions reviewed.

The Standard Model of particle physics is believed to be only the low energy limit of a more fundamental theory. In order to determine its range of validity, a major part of theoretical and experimental efforts in physics is dedicated to precision tests of the Standard Model. Lattice QCD is a non-perturbative, first-principles approach to Quantum Field Theory. It plays an important role in flavor physics by providing calculations of non-perturbative strong interaction contributions to weak processes involving quarks. Measurements of hadronic quantities can be used to constrain the Standard Model as well as theories Beyond the Standard Model. The first part of this thesis contains theoretical developments regarding non-perturbative renormalization. A new renormalization scheme, RI/mSMOM, for fermion bilinear operators in QCD at non-vanishing quark mass is presented. In order to investigate the properties of the mSMOM scheme, an explicit one-loop computation in perturbation theory using dimensional regularization is performed. Numerically, vertex functions are generated on the lattice, with an appropriate projector, based on the RI/SMOM scheme and the renormalization factors are extracted. Quantities measured include renormalization of the axial current ZA, required to renormalize the axial current entering the computation of the decay constant and the renormalization of the bag parameter. The second part of this report focuses on flavor physics phenomenology on the lattice. It presents results of the first run of the RBC/UKQCD charm project with (2+1)-flavor Domain Wall fermions. Observables and matrix elements are measured on lattices with Iwasaki gauge action. There are two ensembles at the physical point with inverse lattice spacings 1.73 and 2.36 GeV and a third finer ensemble at 2.76 GeV as well as four other auxiliary ensembles with smaller volumes and heavier pion masses which are used to perform the continuum extrapolations. The quantities measured in the region of the charm quark mass are meson masses, decay constants, the matrix element of the OV V +AA operator, the neutral D-meson mixing parameter B and the SU(3) breaking ratio ξ.

One of the most central concepts in condensed matter physics is the electronic band structure. Although band theory was established more than 80 years ago, recent developments have led to new insights that are formulated in the framework of topological band theory. In this thesis a subset of topological band theory is presented, with particular focus on topological supercon- ductors and accompanying Majorana fermions. While simple models are used to introduce basic concepts, a physically more realistic model is also studied intensely in the papers. Through self- consistent tight-binding calculations it is confirmed that Majorana fermions appear in vortex cores and at wire end points when the superconductor is in the topologically non-trivial phase. Many other properties such as the topological invariant, experimental signatures in the local density of states and spectral function, unconventional and odd-frequency pairing, the precense of spin-polarized currents and spin-polarization of the Majorana fermions, and a local π-phase shift in the order parameter at magnetic impurities are also investigated.

I traguardi sperimentali raggiunti negli ultimi decenni nell'ambito della fisica degli atomi ultrafreddi hanno reso possibile la realizzazione in laboratorio di sistemi quantistici con potenziali che decadono con la distanza tramite una legge di potenza. In questo ambito di ricerca si inserisce il modello di Kitaev bidimensionale su un reticolo quadrato che descrive un sistema di fermioni spinless. Questo sistema fisico è caratterizzato da un'Hamiltoniana avente un termine di pairing di tipo p-wave che decade con la distanza secondo una legge di potenza. Il potenziale a lungo raggio conferisce propietà peculiari, assenti per sistemi descritti da Hamiltoniane locali, come ad esempio delle funzioni di correlazione che possiedono due regimi, il primo in cui tendono a zero tramite un andamento ibrido, ovvero in modo esponenziale a corte distanze ed in modo algebrico a lunghe distanze, ed il secondo in cui tendono a zero seguendo un andamento puramente algebrico. Inoltre, se il termine di pairing è sufficientemente forte, sono possibili la violazione della legge dell'area dell'entropia di von Neumann anche per fasi non critiche. In questo lavoro di tesi abbiamo caratterizzato le fasi del modello di Kitaev bidimensionale che, essendo descritto da un'Hamiltoniana fermionica quadratica, è diagonalizzabile in maniera esatta. Lo studio delle diverse fasi del sistema è avvenuto tramite l'analisi dello spettro energetico, delle funzioni di correlazione e dello scaling dell'entropia di von Neumann. Questi strumenti di indagine sono stati ottenuti sia tramite risultati analitci sia tramite simulazioni numeriche. Le proprietà ottenute utilizzando questo approccio sono state riassunte in un diagramma di fase posto nell'ultimo capitolo di questo elaborato.

Within condensed matter physics, systems with strong electronic correlations give rise to fascinating phenomena which characteristically require a physical description beyond a one-electron theory, such as high temperature superconductivity, or Mott metal-insulator transitions. In this thesis, a class of strongly correlated electron systems is considered. These systems exhibit fractionally charged excitations with charge +e/2 or -e/2 in two dimensions (2D) and three dimensions (3D), a consequence of both strong correlations and the geometrical frustration of the interactions on the underlying lattices. Such geometrically frustrated systems are typically characterized by a high density of low-lying excitations, leading to various interesting physical effects. This thesis constitutes a study of a model of spinless fermions on the geometrically frustrated kagome lattice. Focus is given in particular to the regime in which nearest-neighbour repulsions V are large in comparison with hopping t between neighbouring sites, the regime in which excitations with fractional charge occur. In the classical limit t = 0, the geometric frustration results in a macroscopically large ground-state degeneracy. This degeneracy is lifted by quantum fluctuations. A low-energy effective Hamiltonian is derived for the spinless fermion model for the case of 1/3 filling in the regime where |t| << V . In this limit, the effective Hamiltonian is given by ring-exchange of order ~ t^3/V^2, lifting the degeneracy. The effective model is shown to be equivalent to a corresponding hard-core bosonic model due to a gauge invariance which removes the fermionic sign problem. The model is furthermore mapped directly to a Quantum Dimer model on the hexagonal lattice. Through the mapping it is determined that the kagome lattice model exhibits plaquette order in the ground state and also that fractional charges within the model are linearly confined. Subsequently a doped version of the effective model is studied, for the case where exactly one spinless fermion is added or subtracted from the system at 1/3 filling. The sign of the newly introduced hopping term is shown to be removable due to a gauge invariance for the case of hole doping. This gauge invariance is a direct result of the bipartite nature of the hole hopping and is confirmed numerically in spectral density calculations. For further understanding of the low-energy physics, a derivation of the model gauge field theory is presented and discussed in relation to the confining quantum electrodynamic in two dimensions. Exact diagonalization calculations illustrate the nature of the fractional charge confinement in terms of the string tension between a bound pair of defects. The calculations employ topological symmetries that exist for the manifold of ground-state configurations. Dynamical calculations of the spectral densities are considered for the full spinless fermion Hamiltonian and compared in the strongly correlated regime with the doped effective Hamiltonian. Calculations for the effective Hamiltonian are then presented for the strongly correlated regime where |t| << V . In the limit g << |t|, the fractional charges are shown to be effectively free in the context of the finite clusters studied. Prominent features of the spectral densities at the Gamma point for the hole and particle contributions are attributed to approximate eigenfunctions of the spinless fermion Hamiltonian in this limit. This is confirmed through an analytical derivation. The case of g ~ t is then considered, as in this case the confinement of the fractional charges is observable in the spectral densities calculated for finite clusters. The bound states for the effectively confined defect pair are qualitatively estimated through the solution of the time-independent Schroedinger equation for a potential which scales linearly with g. The double-peaked feature of spectral density calculations over a range of g values can thus be interpreted as a signature of the confinement of the fractionally charged defect pair. Furthermore, the metal-insulator transition for the effective Hamiltonian is studied for both t > 0 and t < 0. Exact diagonalization calculations are found to be consistent with the predictions of the effective model. Further calculations confirm that the sign of t is rendered inconsequential due to the gauge invariance for g in the regime |t| << V . The charge-order melting metal-insulator transition is studied through density-matrix renormalization group calculations. The opening of the energy gap is found to differ for the two signs of t, reflecting the difference in the band structure at the Fermi level in each case. The qualitative nature of transition in each case is discussed. As a step towards a realization of the model in experiment, density-density correlation functions are introduced and such a calculation is shown for the plaquette phase for the effective model Hamiltonian at 1/3 filling in the absence of defects. Finally, the open problem of statistics of the fractional charges is discussed.

Since the original discovery of heavy fermion behavior in the late seventies by Andres, heavy fermions keep attracting scientific interest due to their exotic and unusual properties. These are inter-metallic compounds that contain rare earth elements, like cerium, praseodymium, and ytterbium, and actinides like uranium. The term ``heavy'' refers to their large effective electronic mass, as large as 1000 times the normal metal ones. The active physics in these materials results from the magnetic moments, associated to the partially filled $f$-shells of rare earth or actinide ions, being immersed into a quantum sea of mobile conduction electrons. In most rare earth metals and insulators, local moments tend to order magnetically, but in heavy electron metals the quantum mechanical jiggling of the local moments induced by delocalized electrons is fierce enough to melt magnetic order. The mechanism by which this takes place involves a remarkable piece of quantum physics known as the ``Kondo effect'' that describes the process by which a magnetic impurity get screened by conduction electrons, forming the so-called Kondo singlet below a characteristic temperature/energy scale named the Kondo temperature, $T_K$. Even though the Kondo effect refers strictly speaking to a very dilute concentration of magnetic ions, typically few part per million, the same physics is believed to play a role in heavy fermions. Heavy fermion materials have become recently popular also in the study of the quantum critical behavior of matter in the vicinity of a zero temperature second-order phase transition. Indeed, heavy fermions realize prototypical examples of quantum critical points that separate at zero temperature magnetic and paramagnetic phases. Experimentally, quantum critical points are attained by tuning non-thermal control parameters, such as pressure, chemical doping or applied magnetic field, so as to drive continuously to zero the magnetic ordering temperature. One of presently lively discussions up to date is about the appearance of two types of magnetic quantum critical points, depending on the behavior of the Kondo singlet as the transition is approached from paramagnetic side. If the Kondo singlet remains across the magnetic transition, the latter is of a spin-density-wave type in which the only critical degrees of freedom are the fluctuations of the magnetic order parameter. In this scenario, the Fermi volume does not change and contains both $f$ and conduction electrons. The alternative scenario invokes instead a local quantum criticality, where the Kondo singlet breaks down right at the magnetic transition, bringing about novel critical modes. Across such a quantum critical point, one expects a sudden collapse of the large Fermi surface of the paramagnetic side to a small magnetic one that contains only conduction electrons. Around a quantum critical point interesting phenomena such as non-Fermi liquid behavior or the appearance of exotic phases may appear. Indeed, many heavy fermions show superconductivity right after the magnetic transition. There are also evidences of coexisting magnetism and superconductivity. Emergence of superconductivity in heavy fermions is at first glance quite surprising, since in the conventional wisdom magnetic impurity scattering is pair-breaking. The evidence of non-$s$ wave symmetry of the order parameter brings these materials in the class of unconventional superconductors, where pairing is not phonon-mediated but likely caused by magnetic fluctuations. This issue has attracted a lot of experimental and theoretical interest. From the theoretical point of view, already building up a microscopic Hamiltonian that could capture the main physics and reproduce the phase diagram of heavy fermions is a challenge that is still ongoing. One of the first attempts to attack this issue was done by Anderson, who proposed in 1961 the model that is nowadays universally known as the Anderson impurity model. Later on, Doniach introduced a lattice version believed to describe heavy fermions, the so-called Kondo lattice model. The latter one has been studied extensively and there is a strong belief that it indeed captures the basic physics of heavy fermions. In this Thesis we study the ground-state phase diagram of various versions of the Kondo lattice model in two dimensions, starting from the simplest Doniach's one, with special focus on the possible appearance of superconductivity in the phase diagram. To attack this problem, we adopt a variational Monte Carlo scheme that allows treating quite large lattices, thus going beyond the one-dimensional and, at the opposite, the infinite-dimensional cases where most of the numerical studies have been restricted so far. Using Gutzwiller projected wave functions we are able to satisfy the local constraint of one electron per $f$ orbital locally not in average: this is the main advantage of the variational Monte Carlo against the mean-field approach. The flexibility of this variational method makes it possible to account for different types of correlations (specially pairing correlations) in the trial wave function, which are not present at the mean-field level. A full optimization of the variational wave function allows us to finally depict the phase diagram.

In dieser Arbeit werden Gross-Neveu Modelle mit einer endlichen Anzahl von Fermiontypen auf einem zweidimensionalen Euklidischen Raumzeitgitter betrachtet. Modelle dieses Typs sind asymptotisch frei und invariant unter einer chiralen Symmetrie. Aufgrund dieser Gemeinsamkeiten mit QCD sind sie sehr gut geeignet als Testumgebungen für Fermionwirkungen die in großangelegten Gitter-QCD-Rechnungen benutzt werden. Das Schrödinger Funktional für die Gross-Neveu Modelle wird definiert für Wilson und Ginsparg-Wilson Fermionen. In 1-Schleifenstörungstheorie wird seine Renormierbarkeit gezeigt. Die Vier-Fermionwechselwirkungen der Gross-Neveu Modelle habe dimensionslose Kopplungskonstanten in zwei Dimensionen. Die Symmetrieeigenschaften der Vier-Fermionwechselwirkungen und deren Beziehungen untereinander werden diskutiert. Im Fall von Wilson Fermionen ist die chirale Symmetrie explizit gebrochen und zusätzliche Terme müssen in die Wirkung aufgenommen werden. Die chirale Symmetrie wird durch das Einstellen der nackten Masse und einer der Kopplungen bis auf Cut-off-Effekte wiederhergestellt. Die kritische Masse und die symmetriewiederherstellende Kopplung werden bis zur zweiten Ordnung in Gitterstörungstheorie berechnet. Dieses Resultat wird in der 1-Schleifenberechnung der renormierten Kopplungen und der zugehörigen Betafunktionen benutzt. Die renormierten Kopplungen werden definiert mit Hilfe von geeignete Rand-Rand-Korrelatoren. Die Rechnung reproduziert die bekannten führenden Koeffizienten der Betafunktionen. Eine der Kopplungen hat eine verschwindende Betafunktion. Die Rechnung wird mit dem vor kurzem vorgeschlagenen Schrödinger Funktional mit exakter chiraler Symmetrie, also Ginsparg Wilson Fermionen, wiederholt. Es werden die gleichen Divergenzen gefunden, wie im Fall von Wilson Fermionen. Unter Benutzung des regularisierungsabhängigen, endlichen Teils der renormierten Kopplungen werden die Verhältnisse der Lambda-Parameter bestimmt.
Gross-Neveu type models with a finite number of fermion flavours are studied on a two-dimensional Euclidean space-time lattice. The models are asymptotically free and are invariant under a chiral symmetry. These similarities to QCD make them perfect benchmark systems for fermion actions used in large scale lattice QCD computations. The Schrödinger functional for the Gross-Neveu models is defined for both, Wilson and Ginsparg-Wilson fermions, and shown to be renormalisable in 1-loop lattice perturbation theory. In two dimensions four fermion interactions of the Gross-Neveu models have dimensionless coupling constants. The symmetry properties of the four fermion interaction terms and the relations among them are discussed. For Wilson fermions chiral symmetry is explicitly broken and additional terms must be included in the action. Chiral symmetry is restored up to cut-off effects by tuning the bare mass and one of the couplings. The critical mass and the symmetry restoring coupling are computed to second order in lattice perturbation theory. This result is used in the 1-loop computation of the renormalised couplings and the associated beta-functions. The renormalised couplings are defined in terms of suitable boundary-to-boundary correlation functions. In the computation the known first order coefficients of the beta-functions are reproduced. One of the couplings is found to have a vanishing beta-function. The calculation is repeated for the recently proposed Schrödinger functional with exact chiral symmetry, i.e. Ginsparg-Wilson fermions. The renormalisation pattern is found to be the same as in the Wilson case. Using the regularisation dependent finite part of the renormalised couplings, the ratio of the Lambda-parameters is computed.

In der vorliegenden Arbeit wird die Physik von Schwer-Fermion-Systemen, die durch Lanthanid- und Aktinid-Übergangsmetallverbindungen realisiert werden, untersucht. Die Basis für eine theoretische Beschreibung bildet das Anderson-Gitter, welches das Wechselspiel freier Leitungselektronen und stark korrelierter Elektronen aus lokalisierten f-Orbitalen charakterisiert. Als Zugang zu diesem Modell wird die von Wegner vorgeschlagene Flussgleichungsmethode verwendet, ein analytisches Verfahren, welches auf der Konstruktion eines effektiven Hamilton-Operators basiert. Ein zentrales Thema dieser Arbeit ist die Beschreibung der elektronischen Struktur von Schwer-Fermion-Systemen. Insbesondere wird die Abhängigkeit statischer Größen vom Einfluss verschiedener Systemparameter betrachtet. Die Dynamik kollektiver Anregungen in Schwer-Fermion-Systemen wird an Hand der elektronischen Zustandsdichten und dynamischen magnetischen Suszeptibilitäten untersucht
The physical properties of heavy-fermion systems are examined. These systems are mainly formed by rare earth or actinide compounds. Their essential physics can be characterized by the periodic Anderson model which describes the interplay of itinerant metal electrons and localized, but strongly correlated f-electrons. The present calculations are based on the flow equations approach proposed by Wegner. This method uses a continuous unitary transformation to derive an effective Hamiltonian of an easy to treat structure. Within this framework the electronic structure of heavy-fermion systems is calculated and the influence of external parameters is studied. Beside the derivation of static properties the density of states and dynamic magnetic susceptibilities are investigated in order to characterize the nature of collective excitations

In der vorliegenden Arbeit wird die Physik von Schwer-Fermion-Systemen, die durch Lanthanid- und Aktinid-Übergangsmetallverbindungen realisiert werden, untersucht. Die Basis für eine theoretische Beschreibung bildet das Anderson-Gitter, welches das Wechselspiel freier Leitungselektronen und stark korrelierter Elektronen aus lokalisierten f-Orbitalen charakterisiert. Als Zugang zu diesem Modell wird die von Wegner vorgeschlagene Flussgleichungsmethode verwendet, ein analytisches Verfahren, welches auf der Konstruktion eines effektiven Hamilton-Operators basiert. Ein zentrales Thema dieser Arbeit ist die Beschreibung der elektronischen Struktur von Schwer-Fermion-Systemen. Insbesondere wird die Abhängigkeit statischer Größen vom Einfluss verschiedener Systemparameter betrachtet. Die Dynamik kollektiver Anregungen in Schwer-Fermion-Systemen wird an Hand der elektronischen Zustandsdichten und dynamischen magnetischen Suszeptibilitäten untersucht.
The physical properties of heavy-fermion systems are examined. These systems are mainly formed by rare earth or actinide compounds. Their essential physics can be characterized by the periodic Anderson model which describes the interplay of itinerant metal electrons and localized, but strongly correlated f-electrons. The present calculations are based on the flow equations approach proposed by Wegner. This method uses a continuous unitary transformation to derive an effective Hamiltonian of an easy to treat structure. Within this framework the electronic structure of heavy-fermion systems is calculated and the influence of external parameters is studied. Beside the derivation of static properties the density of states and dynamic magnetic susceptibilities are investigated in order to characterize the nature of collective excitations.

Les systèmes électroniques bidimensionnels sont d'une grande actualité tout particulièrement depuis la découverte de la supraconductivité à haute température. Ici, on se restreint à l'étude d'un modèle de Hubbard étendu, à la limite d'un couplage faible. En général, le gaz électronique subit une instabilité supraconductrice même sans phonons. Cependant, dans le cas spécial d'une bande demi-remplie, la surface de Fermi est emboîtée et se trouve à une singularité de Van Hove. Cette situation conduit à une compétition entre six instabilités différentes. Outre la supraconductivité en onde $s$ et $d$, on trouve des ondes de densités de spin et de charge ainsi que deux phases qui sont caractérisées par des courants circulaires de charge et de spin respectivement. Le formalisme du groupe de renormalisation est présenté en reliant l'idée de la "< sommation parquet "> au concept plus moderne de l'action effective de Wilson. Comme résultat on obtient un diagramme de phases riche en fonction de l'interaction du modèle. Ce diagramme de phase est exact dans la limite d'une interaction infiniment faible, puisque dans ce cas les lignes de transitions sont fixées par des symétries du modèle. Les comportements à basse température de la susceptibilité de spin ainsi que de la compressibilité de charge complètent l'image physique de ces instabilités. Il s'avère que la surface de Fermi à une tendence générale de se déformer spontanément, mais l'emboîtement n'est pas détruit. En résumé, le modèle de Hubbard à couplage faible reproduit deux propriétés essentielles des cuprates: une phase antiferromagnetique à demi remplissage et la supraconductivité en onde $d$ dans le cas dopé. Mais elle n'éxplique pas les propriétés inhabituelles de l'état métallique dans le régime sous-dopé. Une extension systématique de l'approche perturbative pourrait aider à mieux comprendre ces propriétés, mais reste difficile puisque les techniques nécessaires ne sont pas encore complètement développées.

Neste trabalho estudamos o modelo da rede de Kondo em uma rede kagome, buscando uma maior compreensão dos efeitos da frustração geométrica em sistemas de férmions pesados. Para tanto, fizemos uma aproximação de campo médio no hamiltoniano do sistema que serve para todas as fases do sistema. Analisamos inicialmente o caso não magnético. Obtemos neste limite as energias eletrônicas e as funções de Green necessárias ao cálculo numérico autoconsistente das ocupações e do parâmetro de Kondo. Os resultados encontrados estão em concordância qualitativa com trabalhos publicados em outras geometrias. A seguir analisamos o caso magnético, onde introduzimos uma aproximação suplementar, a qual é compatível com a de campo médio já considerada e, em princípio, existente apenas em sistemas com frustração geométrica. Realizamos cálculos autoconsistentes através de somas sobre as frequências de Matsubara. Os resultados mostram que não há coexistência entre ordem magnética e efeito Kondo, além de haver a supressão do antiferromagnetismo com o aumento de temperatura e variações no preenchimento de bandas.
In this work we study the Kondo Lattice model for the kagome lattice, in order to understand better the effects of geometrical frustration in heavy-fermion systems. In this context, we consider a mean field scheme valid for all the system’s phases. Firstly, we analyzed the nonmagnetic case. In this approximation the electron energies and spectral functions are reachable, then we use the density of states to calculate the occupations selfconsistently. Our results are qualitatively compared with previous works in other geometries. In the second part we introduce an approximation for magnestism, which takes into account the mean field scheme considered and the presence of geometrical frustration. Self-consistent calculations are done through the frequencies summation method. Our results show that the magnetism is supressed when the temperature is increased or the band filling deviates from half-filling. Besides, the coexistence of magnetic order and Kondo effect is not observable.

Etude de la dilatation thermique negative de ceal::(3) associee au comportement de la chaleur specifique sous pression, en utilisant des resultats de methodes variationnelles appliquees au reseau d'anderson. Etude des proprietes de la phase supraconductrice des fermions lourds en relation avec les differentes symetries possibles pour le parametre d'ordre. Calcul de l'effet d'impuretes non magnetiques sur la temperature critique des phases de symetrie s-etendu et d. Etude du reseau d'anderson dans la limite d'une faible amplitude de saut

Lu, Wenlong = 一维光格子中费米型原子混合物基态行为的玻色化研究 / 魯文龙.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.
Includes bibliographical references (p. 70-72).
Abstract also in Chinese.
Lu, Wenlong = Yi wei guang ge zi zhong Feimi xing yuan zi hun he wu ji tai xing wei de Bose hua yan jiu / Lu Wenlong.
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Cold-atom systems --- p.1
Chapter 1.1.1 --- Optical lattices --- p.2
Chapter 1.1.2 --- Feshbach resonance --- p.3
Chapter 1.2 --- Outline of the thesis --- p.6
Chapter 2 --- Bosonization method --- p.8
Chapter 2.1 --- Special property of one-dimensional Fermion system --- p.9
Chapter 2.2 --- Bosonization techniques --- p.13
Chapter 2.2.1 --- Density operators as bosonic fields --- p.14
Chapter 2.2.2 --- Bosonization Identities --- p.17
Chapter 2.3 --- Renormalization analysis for Sine-Gordon field --- p.19
Chapter 2.4 --- Summary --- p.25
Chapter 3 --- Mass imbalance in the spin polarized fermion system --- p.26
Chapter 3.1 --- Kinetic term --- p.29
Chapter 3.2 --- Interaction term --- p.32
Chapter 3.3 --- Phase separation --- p.38
Chapter 3.4 --- Dominant order and pairing behavior --- p.47
Chapter 3.5 --- Summary --- p.49
Chapter 4 --- Mass imbalance in the strong repulsive interaction region --- p.50
Chapter 4.1 --- Effective Hamiltonian at large U limit --- p.50
Chapter 4.2 --- Bosonization of t-J-Jz model --- p.54
Chapter 4.3 --- Phase separation --- p.60
Chapter 4.4 --- Summary --- p.67
Chapter 5 --- Conclusions --- p.68
Bibliography --- p.70
Chapter A --- Proofs of Bosonization --- p.73
Chapter A.1 --- Anti-commutation relations between two branches of fermionic field operators --- p.73
Chapter A.2 --- Bosonization-identities checking --- p.74
Chapter B --- Diagonalization of Quadratic Hamiltonian with Two Bosonic Fields --- p.77
Chapter C --- Correlation functions --- p.82

Ultra-relativisitic heavy ion collisions produce quark gluon plasma-a hot and dense soup of deconfined quarks and gluons akin to the early universe. We study two models in the context of these collisions namely, Polyakov Quark Meson Model (PQM) and Hadron Resonance Gas Model (HRGM).The PQM Model provides us with a simple and intuitive understanding of the QCD equation of state and thermodynamics at non zero temperature and baryon density while the HRGM is the principle model to analyse the hadron yields measured in these experiments across the entire range of beam energies. We study the effect of including the commonly neglected fermionic vacuum fluctuations to the (2+1) flavor PQM model. The conventional PQM model suffers from a rapid phase transition contrary to what is found through lattice simulations. Addition of the vacuum term tames the rapid transition and significantly improves the model’s agreement to lattice data. We further investigate the role of the vacuum term on the phase diagram. The smoothening effect of the vacuum term persists even at non zero . Depending on the value of the mass of the sigma meson, including the vacuum term results in either pushing the critical end point into higher values of the chemical potential or excluding the possibility of a critical end point altogether. We compute the fluctuations(correlations) of conserved charges up to sixth(fourth) order. Comparison is made with lattice data wherever available and overall good qualitative agreement is found, more so for the case of the normalised susceptibilities. The model predictions for the ratio of susceptibilities approach to that of an ideal gas of hadrons as in HRGM at low temperatures while at high temperature the values are close to that of an ideal gas of massless quarks. We examine the stability of HRGMs by extending them to take care of undiscovered resonances through the Hagedorn formula. We find that the influence of unknown resonances on thermodynamics is large but bounded. We model the decays of resonances and investigate the ratios of particle yields in heavy-ion collisions. We find that extending these models do not have much effect on hydrodynamics but the hadron yield ratios show better agreement with experiment. In principle HRGMs are internally consistent up to a temperature higher than the cross over temperature in QCD; but by examining quark number susceptibilities we find that their region of applicability seems to end even below the QCD cross over.

Ultra-relativisitic heavy ion collisions produce quark gluon plasma-a hot and dense soup of deconfined quarks and gluons akin to the early universe. We study two models in the context of these collisions namely, Polyakov Quark Meson Model (PQM) and Hadron Resonance Gas Model (HRGM).The PQM Model provides us with a simple and intuitive understanding of the QCD equation of state and thermodynamics at non zero temperature and baryon density while the HRGM is the principle model to analyse the hadron yields measured in these experiments across the entire range of beam energies. We study the effect of including the commonly neglected fermionic vacuum fluctuations to the (2+1) flavor PQM model. The conventional PQM model suffers from a rapid phase transition contrary to what is found through lattice simulations. Addition of the vacuum term tames the rapid transition and significantly improves the model’s agreement to lattice data. We further investigate the role of the vacuum term on the phase diagram. The smoothening effect of the vacuum term persists even at non zero . Depending on the value of the mass of the sigma meson, including the vacuum term results in either pushing the critical end point into higher values of the chemical potential or excluding the possibility of a critical end point altogether. We compute the fluctuations(correlations) of conserved charges up to sixth(fourth) order. Comparison is made with lattice data wherever available and overall good qualitative agreement is found, more so for the case of the normalised susceptibilities. The model predictions for the ratio of susceptibilities approach to that of an ideal gas of hadrons as in HRGM at low temperatures while at high temperature the values are close to that of an ideal gas of massless quarks. We examine the stability of HRGMs by extending them to take care of undiscovered resonances through the Hagedorn formula. We find that the influence of unknown resonances on thermodynamics is large but bounded. We model the decays of resonances and investigate the ratios of particle yields in heavy-ion collisions. We find that extending these models do not have much effect on hydrodynamics but the hadron yield ratios show better agreement with experiment. In principle HRGMs are internally consistent up to a temperature higher than the cross over temperature in QCD; but by examining quark number susceptibilities we find that their region of applicability seems to end even below the QCD cross over.

The advancement in the eld of cold-atoms has generated a lot of interest in the condensed matter community. Cold-atom experiments can simulate clean, disor-der/impurity free systems very easily. In these systems, we have a control over various parameters like tuning the interaction between particles by the Feshbach resonance, tuning the hopping between lattice sites by laser intensity and so on. As a result, these systems can be used to mimic various theoretical models, which was hindered because of various experimental limitations. Thus, we have an ex-perimental tool in which we can start with a simple theoretical model and later tune the model experimentally and theoretically to simulate the real materials. This will be helpful in studying the physics of the real materials as we can control interactions as well as the impurities can also be taken care of. But the advance-ment in the eld of cold atoms has seen a roadblock for the fermions in optical lattices. The super uid and anti-ferromagnetic phases has not been achieved for fermions in optical lattices due to the \cooling problem" (entropy issues). In this thesis, we have addressed the issue of the \cooling problem" for fermions in optical lattice systems and studied the system with determinant quantum Monte Carlo technique. We start by giving a general idea of cold-atoms and optical lat-tice potentials, and a brief review of the experimental work going on in the cold-atomic systems. Experimental limitations like \fermionic cooling problem" have been discussed in some detail. Then we proposed a bilayer band-insulator model to circumvent the \entropy problem" and simultaneously increasing the transi-tion temperature for fermions in optical lattices. We have studied the attractive Hubbard model, which is the minimal model for fermions in optical lattices. The techniques that we have used to study the model are mean- eld theory, Gaussian uctuation theory and determinant quantum Monte Carlo numerical technique. . Chapter-1 : provides a general introduction to the ultra-cold atoms, optical lattice and Feshbach resonance. In this chapter we have discussed about cold-atom experiments in optical lattice systems. Here, we have brie y discussed the control over various parameters in the experiments. The goal of these experiments is to realize or mimic many many-body Hamiltonians in experiments, which until now was just a theoretical tool to describe various many-body physics. In the end we give a brief idea for introducing disorder in the cold-atom experiments discuss the limitations of these experiments in realizing the \interesting" super uid and anti-ferromagnetic phases of fermionic Hubbard model in optical lattices. Chapter-2 : gives a brief idea of \Determinant Quantum Monte-Carlo" (DQM C) technique that has been used to study these systems. In this chapter we will discuss the DQM C algorithm and the observables that can be calculated. We will discuss certain limitation of the DQM C algorithm like numerical instability and sign problem. We will brie y discuss how sign problem doesn't occur in the model we studied. Chapter-3 : discusses the way by which we can bypass the \cooling problem" (high entropy state) to get a fermionic super uid state in the cold atom experi-ments. In this chapter we propose a model whose idea hinges on a low-entropy band-insulator state, which can be tuned to super uid state by tuning the on-site attractive interaction by Feshbach resonance. We show through Gaussian uctua-tion theory that the critical temperature achieved is much higher in our model as compared to the single-band Hubbard model. Through detailed variational Monte Carlo calculations, we have shown that the super uid state is indeed the most stable ground state and there is no other competing order. In the end we give a proposal for its realization in the ultra-cold atom optical lattice systems. Chapter-4 : discusses the DQM C study of the model proposed in chapter- 3. Here we have studied the various single-particle properties like momentum distribution, double occupancies which can be easily measured in cold-atom ex-periments. We also studied the pair-pair and the density-density correlations in detail through DQM C algorithm and mapped out the full T U phase diagram. We show that the proposed model doesn't favor the charge density wave for the interaction strengths we are interested in. Chapter-5 : gives a brief idea of the e ect of adding an on-site random disorder in our proposed bilayer-Hubbard model. We study the e ect of random disorder on various single-particle properties which can be easily veri ed in cold-atom ex-periments. We studied the suppression of the pair-pair correlations as we increase the disorder strength in our proposed model. We nd that the critical value of the interaction doesn't change in the weak-disorder limit. We estimated the critical disorder strength needed to destroy the super uid state and argued that the tran-sition from the super uid to Bose-glass phase in presence of disorder lies in the universality class of (d + 1) XY model. In the end, we give a schematic U V phase diagram for our system. Chapter-6 : We studied the bilayer attractive Hubbard model in different lattice geometry, the bilayer honeycomb lattice, both in presence and in absence of the on-site random disorder. We discussed how the pair-pair and density-density cor-relations behave in the presence and absence of disorder. Through the finite-size scaling analysis we see the co-existence of the super fluid and the charge density wave order at half- lling. An in nitesimal disorder destroys the CDW order com-pletely while the super uid phase found to be robust against weak-disorder. We estimated the critical interaction strength, the critical temperature and the critical disorder strength through nite-size scaling, and provide a putative phase diagram for the system considered.

The advancement in the eld of cold-atoms has generated a lot of interest in the condensed matter community. Cold-atom experiments can simulate clean, disor-der/impurity free systems very easily. In these systems, we have a control over various parameters like tuning the interaction between particles by the Feshbach resonance, tuning the hopping between lattice sites by laser intensity and so on. As a result, these systems can be used to mimic various theoretical models, which was hindered because of various experimental limitations. Thus, we have an ex-perimental tool in which we can start with a simple theoretical model and later tune the model experimentally and theoretically to simulate the real materials. This will be helpful in studying the physics of the real materials as we can control interactions as well as the impurities can also be taken care of. But the advance-ment in the eld of cold atoms has seen a roadblock for the fermions in optical lattices. The super uid and anti-ferromagnetic phases has not been achieved for fermions in optical lattices due to the \cooling problem" (entropy issues). In this thesis, we have addressed the issue of the \cooling problem" for fermions in optical lattice systems and studied the system with determinant quantum Monte Carlo technique. We start by giving a general idea of cold-atoms and optical lat-tice potentials, and a brief review of the experimental work going on in the cold-atomic systems. Experimental limitations like \fermionic cooling problem" have been discussed in some detail. Then we proposed a bilayer band-insulator model to circumvent the \entropy problem" and simultaneously increasing the transi-tion temperature for fermions in optical lattices. We have studied the attractive Hubbard model, which is the minimal model for fermions in optical lattices. The techniques that we have used to study the model are mean- eld theory, Gaussian uctuation theory and determinant quantum Monte Carlo numerical technique. . Chapter-1 : provides a general introduction to the ultra-cold atoms, optical lattice and Feshbach resonance. In this chapter we have discussed about cold-atom experiments in optical lattice systems. Here, we have brie y discussed the control over various parameters in the experiments. The goal of these experiments is to realize or mimic many many-body Hamiltonians in experiments, which until now was just a theoretical tool to describe various many-body physics. In the end we give a brief idea for introducing disorder in the cold-atom experiments discuss the limitations of these experiments in realizing the \interesting" super uid and anti-ferromagnetic phases of fermionic Hubbard model in optical lattices. Chapter-2 : gives a brief idea of \Determinant Quantum Monte-Carlo" (DQM C) technique that has been used to study these systems. In this chapter we will discuss the DQM C algorithm and the observables that can be calculated. We will discuss certain limitation of the DQM C algorithm like numerical instability and sign problem. We will brie y discuss how sign problem doesn't occur in the model we studied. Chapter-3 : discusses the way by which we can bypass the \cooling problem" (high entropy state) to get a fermionic super uid state in the cold atom experi-ments. In this chapter we propose a model whose idea hinges on a low-entropy band-insulator state, which can be tuned to super uid state by tuning the on-site attractive interaction by Feshbach resonance. We show through Gaussian uctua-tion theory that the critical temperature achieved is much higher in our model as compared to the single-band Hubbard model. Through detailed variational Monte Carlo calculations, we have shown that the super uid state is indeed the most stable ground state and there is no other competing order. In the end we give a proposal for its realization in the ultra-cold atom optical lattice systems. Chapter-4 : discusses the DQM C study of the model proposed in chapter- 3. Here we have studied the various single-particle properties like momentum distribution, double occupancies which can be easily measured in cold-atom ex-periments. We also studied the pair-pair and the density-density correlations in detail through DQM C algorithm and mapped out the full T U phase diagram. We show that the proposed model doesn't favor the charge density wave for the interaction strengths we are interested in. Chapter-5 : gives a brief idea of the e ect of adding an on-site random disorder in our proposed bilayer-Hubbard model. We study the e ect of random disorder on various single-particle properties which can be easily veri ed in cold-atom ex-periments. We studied the suppression of the pair-pair correlations as we increase the disorder strength in our proposed model. We nd that the critical value of the interaction doesn't change in the weak-disorder limit. We estimated the critical disorder strength needed to destroy the super uid state and argued that the tran-sition from the super uid to Bose-glass phase in presence of disorder lies in the universality class of (d + 1) XY model. In the end, we give a schematic U V phase diagram for our system. Chapter-6 : We studied the bilayer attractive Hubbard model in different lattice geometry, the bilayer honeycomb lattice, both in presence and in absence of the on-site random disorder. We discussed how the pair-pair and density-density cor-relations behave in the presence and absence of disorder. Through the finite-size scaling analysis we see the co-existence of the super fluid and the charge density wave order at half- lling. An in nitesimal disorder destroys the CDW order com-pletely while the super uid phase found to be robust against weak-disorder. We estimated the critical interaction strength, the critical temperature and the critical disorder strength through nite-size scaling, and provide a putative phase diagram for the system considered.

Lattice gauge theory is an important approach to understanding quantum chromodynamics (QCD) due to the large coupling constant in the theory at low energy. In this thesis, we report our study of the topological properties of the gauge fields and we calculate 𝘮_η and 𝘮_η' which are related to the topology of the gauge fields. We also develop two algorithms to speed up the inversion of the Dirac equation which is computationally demanding in lattice QCD calculations. The topology of lattice gauge fields is important but difficult to study because of the large local fluctuations of the gauge fields. In chapter 2, we probe the topological properties of the gauge fields through the measurement of closed quark loops, field strength and low-lying eigenvectors of the Shamir domain wall operator. The closed quark loops suggest the slow evolution of topological modes during the generation of QCD configurations. The chirality of the low-lying eigenvectors is studied and the lattice eigenvectors are compared to the eigenvectors in the continuous theory. The topological charges are calculated from the eigenvectors and the results agree with the topological charges calculated from the smoothed gauge fields. The fermion correlators are also obtained from the eigenvectors. The non-trivial topological properties of QCD gauge fields are important to the mass of the η and η', 𝘮_η and 𝘮_η'. Lattice QCD is an area where 𝘮_{\eta}$ and 𝘮_{\eta'}$ can be calculated by using gauge fields that are sampled over different topological sectors. We calculate 𝘮_η and 𝘮_η' in chapter 3 by including the fermion correlators and the topological charge density correlators. The errors of 𝘮_η and 𝘮_η' are reduced to the percent level and the mixing angle between the octet, singlet states in the SU(3) limit and the physical eigenstates is calculated. An algorithm that reduces communication and increases the usage of the local computational power is developed in chapter 4. The algorithm uses the multisplitting algorithm as a preconditioner in the preconditioned conjugate gradient method. It speeds up the inversion of the Dirac equation during the evolution phase. In chapter 5, we utilize two lattices, called the coarse lattice and the fine lattice, that lie on the renormalization group trajectory and have different lattice spacings. We find that the low-mode space of the coarse lattice corresponds to the low-mode space of the fine lattice. Because of the correspondence, the coarse lattice can be used to solve the low modes of the fine lattice. The coarse lattice is used in the restart algorithm and the preconditioned conjugate gradient algorithm where the latter is called the renormalization group based preconditioned conjugate gradient algorithm (RGPCG). By using the near-null vectors as the filter, RGPCG could reduce the operations of the matrix multiplications on the fine lattice by 33% to 44% for the inversion of Dirac equation. The algorithm works better than the conjugate gradient algorithm when multiple equations are solved.

Within condensed matter physics, systems with strong electronic correlations give rise to fascinating phenomena which characteristically require a physical description beyond a one-electron theory, such as high temperature superconductivity, or Mott metal-insulator transitions. In this thesis, a class of strongly correlated electron systems is considered. These systems exhibit fractionally charged excitations with charge +e/2 or -e/2 in two dimensions (2D) and three dimensions (3D), a consequence of both strong correlations and the geometrical frustration of the interactions on the underlying lattices. Such geometrically frustrated systems are typically characterized by a high density of low-lying excitations, leading to various interesting physical effects. This thesis constitutes a study of a model of spinless fermions on the geometrically frustrated kagome lattice. Focus is given in particular to the regime in which nearest-neighbour repulsions V are large in comparison with hopping t between neighbouring sites, the regime in which excitations with fractional charge occur. In the classical limit t = 0, the geometric frustration results in a macroscopically large ground-state degeneracy. This degeneracy is lifted by quantum fluctuations. A low-energy effective Hamiltonian is derived for the spinless fermion model for the case of 1/3 filling in the regime where |t| << V . In this limit, the effective Hamiltonian is given by ring-exchange of order ~ t^3/V^2, lifting the degeneracy. The effective model is shown to be equivalent to a corresponding hard-core bosonic model due to a gauge invariance which removes the fermionic sign problem. The model is furthermore mapped directly to a Quantum Dimer model on the hexagonal lattice. Through the mapping it is determined that the kagome lattice model exhibits plaquette order in the ground state and also that fractional charges within the model are linearly confined. Subsequently a doped version of the effective model is studied, for the case where exactly one spinless fermion is added or subtracted from the system at 1/3 filling. The sign of the newly introduced hopping term is shown to be removable due to a gauge invariance for the case of hole doping. This gauge invariance is a direct result of the bipartite nature of the hole hopping and is confirmed numerically in spectral density calculations. For further understanding of the low-energy physics, a derivation of the model gauge field theory is presented and discussed in relation to the confining quantum electrodynamic in two dimensions. Exact diagonalization calculations illustrate the nature of the fractional charge confinement in terms of the string tension between a bound pair of defects. The calculations employ topological symmetries that exist for the manifold of ground-state configurations. Dynamical calculations of the spectral densities are considered for the full spinless fermion Hamiltonian and compared in the strongly correlated regime with the doped effective Hamiltonian. Calculations for the effective Hamiltonian are then presented for the strongly correlated regime where |t| << V . In the limit g << |t|, the fractional charges are shown to be effectively free in the context of the finite clusters studied. Prominent features of the spectral densities at the Gamma point for the hole and particle contributions are attributed to approximate eigenfunctions of the spinless fermion Hamiltonian in this limit. This is confirmed through an analytical derivation. The case of g ~ t is then considered, as in this case the confinement of the fractional charges is observable in the spectral densities calculated for finite clusters. The bound states for the effectively confined defect pair are qualitatively estimated through the solution of the time-independent Schroedinger equation for a potential which scales linearly with g. The double-peaked feature of spectral density calculations over a range of g values can thus be interpreted as a signature of the confinement of the fractionally charged defect pair. Furthermore, the metal-insulator transition for the effective Hamiltonian is studied for both t > 0 and t < 0. Exact diagonalization calculations are found to be consistent with the predictions of the effective model. Further calculations confirm that the sign of t is rendered inconsequential due to the gauge invariance for g in the regime |t| << V . The charge-order melting metal-insulator transition is studied through density-matrix renormalization group calculations. The opening of the energy gap is found to differ for the two signs of t, reflecting the difference in the band structure at the Fermi level in each case. The qualitative nature of transition in each case is discussed. As a step towards a realization of the model in experiment, density-density correlation functions are introduced and such a calculation is shown for the plaquette phase for the effective model Hamiltonian at 1/3 filling in the absence of defects. Finally, the open problem of statistics of the fractional charges is discussed.