This PhD thesis is concerned with the investigation of Extended Hubbard Models on one-dimensional lattices (chains); the approach used to such investigation is that of determining exact results. The extended Hubbard models are models of correlated electrons on lattices; they are the extensions of a model proposed by Hubbard in \\cite{HUB} to explain the metal-insulator transition in
d-transition compounds as the effect of electronic correlations, usually neglected in band theories. Later on, a need for extensions of such model was first felt within the context of high-temperature superconductivity; indeed, the different behaviour of doped cuprates with respect to conventional superconductors suggested that a microscopic theory of this phenomenon could be formulated in terms of electron-electron interaction, without explicitly invoking the phonon coupling, like in BCS theory. Although an exhaustive microscopic formulation for cuprate superconductivity has not been achieved yet, the Extended Hubbard Models have been in the meanwhile applied to other problems in condensed matter; they are actually considered the reference paradigm for treating all those systems in which electronic correlations cannot be neglected. \\\\As observed above, this thesis focuses in particular on 1-D chains: indeed in last years the progress in technological synthesization and small-scale observations has shown that there are many classes of materials exhibiting a strong one-dimensional character, such as organic (super)conductors, linear-chain polymers, quantum wires, quantum-dot chains, one-dimensional cuprates. Many experimental results confirm that such materials exhibit a quite rich behaviour, with different phases, charge-spin separation, as well as a strong electronic correlations. However, a sufficient understanding of these phenomena on a theoretical point of view has not been reached yet. This is due to the fact the usual theoretical methods can be shown to fail in D=1, and theoreticians are thus forced to develop new schemes for approaching the problem of many interacting fermions in such dimension. The latter therefore represents a fascinating challenge. Some progress has been made and many aspects are nowadays more clear; nevertheless, due to the lack of traditional techniques, the need for a comparison with exact results is perhaps more urgent in D=1 than in higher dimensions. The aim of this thesis is to provide some results to the research in this field. The present work is organized as follows: the two first chapters are introductory, while chapters 3 and 4 mostly contain the author's original contribution. More explicitly, {\\bf chapter 1} is an introduction to the Hubbard models (both the ordinary and the Extended) and is meant for the reader who is non-familiar with these models. We tried to emphasize the motivation that historically led to their introduction and the physical contexts in which they are applied, not necessarily in one-dimension. \\\\{\\bf Chapter 2} is instead devoted to general problematics of one dimensional physics. We have tried to stress that there is a number of physically interesting materials that can be regarded as actual 1-D systems; we give a brief account of the difficulties leading the usual theoretical approaches, such as mean-field and Fermi Liquid, to fail in D=1. While briefly describing the important results of the proposed alternative formulations (Luttinger Liquid, Renormalization Group), we also point out the points that are still unsatisfactory, and emphasize the importance that exactly integrable models have in this field. \\\\Matching the subjects of chapter 1 and 2, the subsequent ones contain the results of our investigation of exactly solvable Extended Hubbard model on 1-D chains. \\\\We wish to emphasize to the reader non familiar with integrable systems that the literature on this subject is often strongly concerned with the mathematical aspects of the involved methods, leaving the determination of the physical features of the derived integrable models as a secondary issue. On the contrary, we believe that both aspects of the research deserve a relevant role in the study. For these reasons, while chapter 3 contains new results about the general formulation of integrability for fermionic systems, chapter 4 is devoted to the investigation of the physical features for some models whose integrability has been proved in the previous chapter. \\\\More explicitly, in {\\bf chapter 3} (apart from a preliminary introduction of sec.3.1) the theory of integrable {\\it fermionic} systems has been developed within the frame of the Quantum Inverse Scattering Method (QISM). The latter is a technique for obtaining integrable models, traditionally formulated with success for spin systems; its more recent application to electron systems was leaving many unsatisfactory aspects. To overcome this weaknesses, in sec.3.2 we both clarify the theory of matrix representation for fermionic operators, and describe a method for obtaining integrable models of fermions (straightforwardly readable in terms of fermion operators), starting from the the solutions of a
mathbbC-number equation (known as Yang-Baxter Equation). In sec.3.3 we first work out a general method of search for solutions of such equation; secondly we explicitly find 96 solutions, corresponding to 96 integrable Extended Hubbard models (most of which are new) for which a set of mutually commuting operators is proved to exist. In sec.3.4 we formulate a general theorem concerning the symmetries of the integrable models in QISM, and widely discuss the symmetries and supersymmetries for the class of extended Hubbard Models.\\\\ {\\bf Chapter 4} is devoted to the investigation of the physical properties of the integrable Extended Hubbard models found in chapter 3. We introduce a technique, called Sutherland's species technique, that, when applicable, yields an effective reduction of the Hilbert space and considerably simplifies the problem of diagonalizing the Hamiltonian. In fact, such technique is proved to be applicable to all the 96 models found above, and is used to determine the exact ground state phase diagram and the thermodynamics of the models. In particular we both recover in a quite straightforward way the phase diagram of a known model, and that of two new models (Model 2 and Model 3) that were not known before. \\\\The phase diagram exhibit structured forms with respect to the filling (density of electrons) and the on-site Coulomb repulsion (
U); the pressure and the compressibility in the ground state are also calculated. Model 2 exhibits a superconducting phase particularly enhanced towards positive values of
U, as well as a band-controlled insulator-superconductor transition at half filling; model 3 is the first example of exactly solved model with an asymmetric shape with respect to half-filling, due to the absence of particle-hole invariance of the Hamiltonian; a filling controlled metal-insulator transition is found to occur for this model. Generalizations to the other cases are also discussed. \\\\As to the thermodynamics, we compute the exact partition function of Model 3, which we believe to be particular worth of interest due to the above asymmetry. We then derive the exact specific heat of the model as a function of the temperature, which exhibits an interesting two-peak behaviour as well as a nearly universal crossing point. These results are discussed and compared with similar ones present in the literature. The chapters are of course linked each to another according to the above scheme; nevertheless, it has been a purpose of the author to make them as independently readable as possible. In particular, the reader who is more interested in the physical aspects of the models rather than in the general formalism of fermionic QISM can skip chapter 3 (at least in a first-sight reading) and pass directly to chapter 4; apart from some references to previous formulas the understanding should not be affected.\\\\{\\it \\noindent \\underline{Notational Remark:}} The customary symbol to denote the electron filling is
n; this has in fact been used throughout the two first introductory chapters. Nevertheless, in order to avoid possible misunderstanding with other extensive quantities, in chapter 3~and~4~the~symbol~has been changed into
rhoWe present, in this dissertation, a numerical simulation method to study interacting fermion systems. The general simulation procedures are discussed in connection with a description of a Quantum Monte Carlo simulation algorithm for interacting electrons on lattices. The algorithm presented here has been used to simulate interacting electrons on lattices, and it makes possible the study of substantially larger systems than can be studied by other numerical methods. As long as certain limits of applicability are respected, model Hamiltonians of interacting electrons can be studied without resort to uncontrolled approximations. The method then provides nearly exact solutions to model Hamiltonians of many-body systems, in the sense that the degree of error can be controlled. We also discuss some results obtained from simulations of the extended Hubbard model and and the periodic Anderson model. For the extended Hubbard model in two dimensions, different regions are identified where electron correlation produces antiferromagnetism, charge-density-waves, and singlet pairing superconducting behaviors. We also find regions where transitions from one type of correlation to another occur. For the symmetric periodic Anderson model, we observe the formation of local spin moments at high temperatures and their quenching at low temperatures, as in the single impurity Anderson model. In addition, we find antiferromagnetic interaction between f local moments at low temperatures which are not present in a single impurity problem.
"We have studied the extended Hubbard and the Peierls-extended Hubbard models in one dimension using primarily computational methods. We discuss the derivation of the models and the computational methods used to study them, focusing primarily on our extensions over existing numerical methods. In particular, we present the construction of ""operator-loop"" updates for spin-1/2 fermions and the inclusion of electron-phonon interactions in the Stochastic Series Expansion Quantum Monte Carlo method and the use of Quantum Parallel Tempering to improve the quality of the numerical data. For the extended Hubbard model at half-filling, we have confirmed the existence of a recently reported bond-order wave phase for the ground state in a small region of parameter space between the well-known charge- and spin-density wave phases. We have studied in detail the nature of the associated quantum phase transitions between all the phases. Upon doping away from half-filling, the doubly-degenerate, spin and charge gapped long-range-ordered ground states of the model are shown to evolve to a Luther-Emery liquid with no charge gap but a finite spin gap. For the Peierls-extended Hubbard model, which includes both electron-phonon and electron-electron interactions, we have studied the dimerization transition of the half-filled system in the presence of fully quantum mechanical phonons. We show that the transition to a dimerized state occurs only when the electron-phonon coupling exceeds a finite critical value that depends on the bare phonon frequency. This is in contrast to previous studies in similar models with adiabatic phonons that had predicted a dimerized state for arbitrarily small values of the electron-phonon coupling."
We investigate numerically the ground state phase diagram of the one-dimensional extended Hubbard model, including an on-site interaction U and a nearest-neighbor interaction V. We focus on the ground state phases of the model in the $V\ensuremath{\gg}U$ region, where previous studies have suggested the possibility of dominant superconducting pairing fluctuations before the system phase separates at a critical value
V=VPS. Using quantum Monte Carlo methods on lattices much larger than in previous Lancz\"os diagonalization studies, we determine the boundary of phase separation, the Luttinger-liquid correlation exponent ${K}_{\ensuremath{\rho}},$ and other correlation functions in this region. We find that phase separation occurs for V significantly smaller than previously reported. In addition, for negative U, we find that a uniform state reenters from phase separation as the electron density is increased towards half filling. For
V<VPS, our results show that superconducting fluctuations are not dominant. The system behaves asymptotically as a Luttinger liquid with ${K}_{\ensuremath{\rho}}<1,$ but we also find strong low-energy (but gapped) charge-density fluctuations at a momentum not expected for a standard Luttinger liquid.
Numerical results obtained from Monte Carlo simulation in the ground state and at finite temperatures as well as the exact diagonalization and the transfer matrix method are reported. Efficiency of the Monte Carlo method in the ground state is examined. Spin, charge and superconducting correlations are investigated for the Hubbard and the t - J model in one and two dimensions. The momentum distribution in the one-dimensional Hubbard model shows fermi-liquid-like behavior at least in two orders of magnitude smaller energy scale than the band width. The short-range incommensurate spin correlation is observed and analyzed in the doped Hubbard and the t - J model both in one and two dimensions. In the ground state, the superconducting correlation shows the absence of system size dependence in two dimensions at the filling smaller than 0.8 and in one dimension at any filling. The binding energy of two fermions and the spin and charge distortion around an itinerant fermion are also discussed in the t - J and the coupled spin-fermion model.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.