Introduction In the preceding chapters we have developed a number of series expansion methods, and shown how to use these to study lattice spin models of various kinds. While this encompasses a great deal of interesting physics it does not include perhaps the richest, most interesting, and most challenging area of all: strongly correlated electron systems. Of particular interest here are the cuprate high T c superconductors and the manganite ‘colossal magnetoresistance’ (CMR) materials. These systems are far from fully understood but it seems clear that a complex and subtle interplay between charge, spin, and perhaps other degrees of freedom plays a vital role. A number of generic lattice models for strongly correlated electron systems exist. Besides the Hubbard model, which was briefly introduced in Chapter 1, there is its derivative the ‘ t –J model’, the Anderson model, the Kondo lattice model, the Falicov–Kimball model, and others. These all present difficult problems, with little in the way of exact or rigorous results. Many approximate approaches, both analytical and numerical, have been used. In this chapter we will discuss the application of series methods, both at T = 0 and at finite temperatures, to some of these models. Some successes have been achieved but, generally speaking, the series are shorter and the analysis more problematic than for spin models. A particular difficulty, for ground-state calculations, is that series are not well suited to studying continuously variable electron density, although we will illustrate one attempt to overcome this.
Since 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.
Since the genesis of condensed matter physics, strongly correlated fermionic systems have shown a variety of fascinating properties and remain a vital topic in the field. Such systems arise through electronic interaction, and despite decades of intensive research, no holistic approach to solving this problem has been found. During that time, physicists have compiled a wealth of individual experimental and theoretical results, which together give an invaluable insight into these materials, and, in some instances, can explain correlated phenomena. However, there are several systems that stubbornly refuse to fall completely in line with current theoretical descriptions, among them the high-
Tc cuprates and heavy fermion compounds. Although the two material classes have been around for the better part of the last 50 years, large portions of their respective phase diagram are still under intensive debate. Recent experiments in several electron-doped cuprates compounds, e.g. neodymium cerium copper oxide (Nd
2xCe
xCuO
4), reveal a charge ordering about an antiferromagnetic ground state. So far, it has not been conclusively clarified how this intertwining of charge and spin polarization comes about and how it can be reconciled with a rigorous theoretical description. The heavy-fermion semimetals, on the other hand, have enjoyed renewed scientific interest with the discovery of topological Kondo insulators, a new material class offering a unique interface of topology, symmetry breaking, and correlated phenomena. In this context, samarium hexaboride (SmB
6) has emerged as a prototypical system, which may feature a topological ground state. In this thesis, we present a spin rotational invariant auxiliary particle approach to investigate the propensities of interacting electrons towards forming new states of order. In particular, we study the onset of spin and charge order in high-
Tc cuprate systems and Kondo lattices, as well as the interplay of magnetism and topology. To that end, we use a sophisticated mean-field approximation of bosonic auxiliary particles augmented by a stability analysis of the saddle point via Gaussian fluctuations. The latter enables the derivation of dynamic susceptibilities, which describe the response of the system under external fields and offer a direct comparison to experiments. Both the mean-field and fluctuation formalisms require a numerical tool that is capable of extremizing the saddle point equations, on the one hand, and reliably solving a loop integral of the susceptibility-type, on the other. A full, from scratch derivation of the formalism tailored towards a software implementation, is provided and pedagogically reviewed. The auxiliary particle method allows for a rigorous description of incommensurate magnetic order and compares well to other established numerical and analytical techniques. Within our analysis, we employ the two-dimensional one-band Hubbard as well as the periodic Anderson model as minimal Hamiltonians for the high-
Tc cuprates and Kondo systems, respectively. For the former, we observe a regime of intertwined charge- and spin-order in the electron-doped regime, which matches recent experimental observations in the cuprate material Nd
2xCe
xCuO
4. Furthermore, we localize the emergence of a Kondo regime in the periodic Anderson model and establish the magnetic phase diagram of the two-band model for topological Kondo insulators. The emerging antiferromagnetic ground state can be characterized by its topological properties and shows, for a non-trivial phase, topologically protected hinge modes.
A variety of quantum many-body methods have been developed for studying the strongly correlated electron systems. We have also proposed a computationally efficient and accurate approach, named the correlation matrix renormalization (CMR) method to address the challenges. The theoretic development and benchmark tests of the CMR method are included in this thesis. Meanwhile, ground state total energy is the most important property of electronic calculations. We also investigated an alternative approach to calculate the total energy, and extended this method for magnetic anisotropy energy of ferromagnetic materials. In addition, another theoretical tool, dynamical mean-field theory on top of the density functional theory, has also been used in electronic structure calculations for an Iridium oxide to study the phase transition, which results from an interplay of the d electrons' internal degrees of freedom.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.
Relative to single-band models, multiband models of strongly interacting electron systems are of growing interest because of their wider range of novel phenomena and their closer match to the electronic structure of real materials. In this brief review we discuss the physics of three multiband models (the three-band Hubbard, the periodic Anderson, and the Falicov-Kimball models) that was obtained by numerical simulations at zero temperature. We first give heuristic descriptions of the three principal numerical methods (the Lanczos, the density matrix renormalization group, and the constrained-path Monte Carlo methods). We then present generalized versions of the models and discuss the measurables most often associated with them. Finally, we summarize the results of their ground state numerical studies. While each model was developed to study specific phenomena, unexpected phenomena, usually of a subtle quantum mechanical nature, are often exhibited. Just as often, the predictions of the numerical simulations differ from those of mean-field theories.
The discovery of high-Tc superconductivity in cuprates in 1986 moved strongly correlated systems from exotic worlds interesting only for pure theorists to the focus of solid-state research. In recent decades, the majority of hot topics in condensed matter physics (high-Tc superconductivity, colossal magnetoresistance, multiferroicity, ferromagnetism in diluted magnetic semiconductors, etc.) have been related to strongly correlated transition metal compounds. The highly successful electronic structure calculations based on density functional theory lose their predictive power when applied to such compounds. It is necessary to go beyond the mean field approximation and use the many-body theory. The methods and models that were developed for the description of strongly correlated systems are reviewed together with the examples of response function calculations that are needed for the interpretation of experimental information (inelastic neutron scattering, optical conductivity, resonant inelastic X-ray scattering, electron energy loss spectroscopy, angle-resolved photoemission, electron spin resonance, and magnetic and magnetoelectric properties). The peculiarities of (quasi-) 0-, 1-, 2-, and 3- dimensional systems are discussed.