This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus delocalization, which can result in complex and exotic physical phenomena. Strongly correlated fermion systems are usually modelled by many-body Hamiltonians for which the kinetic- and interaction energy have the same order of magnitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives unreliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors. A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to computations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of interacting fermions in two spatial dimensions are also derived and studied.
The primary aim of this thesis is to understand the effects of electronic frustration generated by local interactions of the Hubbard U type on a lattice. Specifically we attempt to study the strongly interacting liquid phase of the Hubbard model and the so called extremely correlated phase of the t-J model without any broken symmetries. Of particular interest is the fate of the Fermi Liquid state under the influence of a large U. Exact solutions in solvable limits inform most of the physical intuition. Perturbative methods such as an infinite order calculation in the particle-particle channel ladder expansion yield fruitful results. Finally the non-perturbative field theory of Schwinger is used to generate a calculation scheme for the Green's function of the non-canonical Fermions of the t-J model. Numerical results are compared with important experimental results from the high-Tc Cuprate compounds.
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
rhoSince the advent of high-
Tc cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent
α for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report
α≳1, which would imply
Kρ≲0.17, a value which cannot be reconciled with the bare Hubbard model where
KρH≥0.5, i.e.,
αH≲1/8. We develop new accurate numerical methods to obtain
Kρ and investigate how to obtain such small values for
Kρ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate
Uc. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.
In this thesis, we discuss quantum phase transitions in low-dimensional optical lattices, namely one- and two-dimensional lattices. The dimensional confinement is realized in experiments by suppressing the hopping in the extra dimensions through a deep potential barrier that prevents the atoms to tunnel across dimensions. The dimensionality considerably affects the physical behavior of the system. For instance, in one dimension, the celebrated Fermi-liquid theory does not describe the many-body behavior of a fermionic system. There are no single particle excitations, but instead collective behavior, and spin and charge degrees of freedom are decoupled . In the first part of this thesis, we consider one-dimensional fermionic gases under the effect of time-dependent driving forces. First, we describe a proposal to realize non-standard Hubbard models that display correlated-hopping terms by modulating interactions through a time-dependent magnetic field. The extra terms describe hopping processes that depend on the particle-density difference in neighboring sites. The effective model is obtained in arbitrary spatial dimensions, but only in one-dimension it is known to be exactly solvable for specific values of the parameters. Moreover, an SU(2) symmetry in the charge sector allows to construct non-trivial superconducting eigenstates (eta-states), discovered by Yang as excited states of the Hubbard model. Then, we study a one-dimensional bipartite optical lattice with fermionic atoms at half-filling. The lattice potential barriers have heights alternating in magnitude. As a consequence, the hopping amplitudes also alternate in magnitude and the elementary unit cell has two sites. This system is known to display Peierls dimerization at half-filling: the fermions form dimers in neighboring sites that inhibit transport and open a gap in the spectrum. After applying shaking on this system we discover that one can dynamically induce a phase transition to an unconventional metallic state with four Fermi-points. The second part of this thesis describes bosonic systems in two-dimensional optical lattices. First, we consider a square optical lattice with alternating deep-shallow wells arranged in a checkerboard pattern. This lattice is realized in the experimental laboratory of Prof. Hemmerich in Hamburg. In the experiment, the visibility is extracted from the time-of-flight imaging to quantify the phase coherence of the bosonic gas. We develop a mean-field theory and determine the phase diagram of this model. We interpret the experimental data by assuming that at large detuning, only the deeper wells are populated and form an imbalanced Mott insulator with empty shallow wells and integer density on the deeper wells. We support this hypothesis by large-scale Gutzwiller calculations. Then, we investigate a system of bosons in the Lieb lattice, which is the lattice geometry of the Cu-O plane in cuprates . We consider configurations of the optical potential where s and p orbitals sitting on neighboring wells become resonant and hybridize. The interest in this problem comes from the possibility to realize time-reversal broken phases, usually known as Varma phases for electronic systems, which have been proposed to explain the pseudo-gap regime of cuprate superconductors. We show the conditions under which the symmetry breaking in a four-band model can lead to states with loop currents. Moreover, the lowest branch of the excitation spectrum has a non-vanishing Chern number and therefore supports topological excitations.
The theoretical description of fermionic system with strong interaction is a very challenging open problem in physics. The most notable (but far from the only) experimental realization of this type of systems are the cuprate superconductors which have zero electric resistivity. Even if onehas a good microscopic model for the description of these materials it is very hard to translate it to macroscopic observables which in principle can be experimentally checked. The problem is that in case of a relevant interaction one can not Taylor expand in the coupling constant in the low-energy regime in which we are most interested. On the other hand, because of the fermion sign problem Monte Carlo numerical techniques (which are succesful with bosonic models) do not work for fermions at finite density. This thesis is devoted to the applications of several methods to the research area described above. The common theme of these techniques is that they are (partly) motivated from high-energy physics: the research area which deals with particle physics, string theory etc.