In this thesis we are going to study the properties of strongly correlated electron systems (SCES) using analytical, phenomenological and computational methods. In the introductory chapter we will discuss generally the problem of strongly correlated electron systems and the methods that are used to analyse their behavior. In chapter 2, we discuss the derivation of low energy effective theories by using perturbation theory. In chapter 3, we point out potential problems with perturbative techniques used to construct low energy theories for SCES. In chapter 4, we introduce an important class of materials, namely the cobaltates NaxCoO2 and we discuss the experiments that establish their strongly correlated electron nature. The cobaltates exhibit a novel insulating state at x = 0.5 which corresponds to a 3/4 filled band. This state is analyzed phenomenologically in chapter 5. In chapter 6, we introduce a non perturbative technique appropriate for strongly correlated electron systems, the Dynamical Mean Field Theory (DMFT) and its cluster extension. The results of the application of cluster DMFT on the cobaltates are presented in chapter 7.
In this thesis we are going to study the properties of strongly correlated electron systems (SCES) using analytical, phenomenological and computational methods. In the introductory chapter we will discuss generally the problem of strongly correlated electron systems and the methods that are used to analyse their behavior. In chapter 2, we discuss the derivation of low energy effective theories by using perturbation theory. In chapter 3, we point out potential problems with perturbative techniques used to construct low energy theories for SCES. In chapter 4, we introduce an important class of materials, namely the cobaltates NaxCoO2 and we discuss the experiments that establish their strongly correlated electron nature. The cobaltates exhibit a novel insulating state at x = 0.5 which corresponds to a 3/4 filled band. This state is analyzed phenomenologically in chapter 5. In chapter 6, we introduce a non perturbative technique appropriate for strongly correlated electron systems, the Dynamical Mean Field Theory (DMFT) and its cluster extension. The results of the application of cluster DMFT on the cobaltates are presented in chapter 7.
In this thesis we study the strongly-correlated-electron physics of the longstanding H-Tc-superconductivity problem using a non-perturbative method, the Dynamical Mean Field Theory (DMFT), capable to go beyond standard perturbation-theory techniques. DMFT is by construction a local theory which neglects spatial correlation. Experiments have however shown that the latter is a fundamental property of cuprate materials. In a first step, we approach the problem of spatial correlation in the normal state of cuprate materials using a phenomenological Fermi-Liquid-Boltzmann model. We then introduce and develop in detail an extension to DMFT, the Cellular Dynamical Mean Field Theory (CDMFT), capable of considering short-ranged spatial correlation in a system, and we implement it using the exact diagonalization algorithm . After benchmarking CDMFT with the exact one-dimensional solution of the Hubbard Model, we employ it to study the density-driven Mott metal-insulator transition in the two-dimensional Hubbard Model, focusing in particular on the anomalous properties of the doped normal state close to the Mott insulator. We finally study the superconducting state. We show that within CDMFT the one-band Hubbard Model supports a d-wave superconductive state, which strongly departs from the standard BCS theory. We conjecture a link between the instabilities found in the normal state and the onset of superconductivity.
Dynamical Mean-Field Theory (DMFT) has opened new perspectives for the investigation of strongly correlated electron systems and greatly improved our understanding of correlation effects in models and materials. In contrast to Hartree-Fock-type approximations the mean field of DMFT is dynamical, whereby local quantum fluctuations are fully taken into account. DMFT becomes exact in the limit of high spatial dimensions or coordination number. Using DMFT the dynamics of correlated electron systems can be investigated non-perturbatively at all interaction strengths, electron densities and temperatures. By merging density functional theory with DMFT a powerful method for the calculation of the properties of correlated electron materials has become available, which is applicable to bulk systems and heterostructures, including topological states of matter. The inclusion of non-local correlations into DMFT makes it possible to explore unconventional superconductivity and the critical behavior at thermal or quantum phase transitions. By generalizing DMFT to non-equilibrium states also the real-time dynamics of correlated systems can be investigated. In this brief review the foundations and current status of DMFT are discussed along with characteristic physical insights obtained with this approach.
In this thesis, several aspects of correlation effects in band insulators that were hitherto not addressed, both in equilibrium and non-equilibrium, are studied using the Ionic Hubbard model (IHM) at half-filling and Dynamical mean-field theory (DMFT). The IHM is an extension of the Hubbard model on a bipartite lattice with a staggered on-site “ionic potential Δ added in. The Hubbard model by itself is a tight-binding model of electrons hopping between (orbitals localized on) lattice sites, with amplitudes t and t’ for first neighbor and second neighbor hopping respectively, together with a coulomb repulsion energy cost of U for double occupancy (i.e., with both up and down spin electrons) at any lattice site. The studies in this thesis address strong correlation effects in this model using DMFT techniques which map the lattice problem to a self consistently embedded quantum impurity problem. The impurity problem itself is solved using state of the art continuous time quantum monte-carlo (CTQMC) techniques (Ch.s 3 and 4) as well as the somewhat simpler Iterated Perturbation Theory (IPT) (Ch.s 3, 5 and 6).
Chapter 3 of the thesis is devoted to a detailed analysis of the properties of the half-filled t-U-Δ IHM (i.e., with t’=0) on a Bethe lattice of infinite connectivity. It is shown conclusively here that for a finite Δ and at zero temperature (T=0), the IHM has a first order transition from the Band Insulator (BI) phase to an antiferromagnetic insulating (AFI) phase at a threshold U = UAF (which increases with Δ) such that the BI to AFI transition preempts the transition from the BI to the correlation induced paramagnetic metallic (PM) phase. Evidence is also presented for a quantum transition to a sliver of half-metal (HM) phase just after the AF order turns on, followed by a return to the AFI phase on further increasing U. With increasing T, the AF order is lost via a first-order transition for weak to intermediate U/t. In the strongly correlated regime, where the effective low-energy Hamiltonian is the Heisenberg model, the transition from the AFI to the paramagnetic Mott insulating phase (adiabatically connected to the BI phase) is of second order. As T increases, the range of U over which the AF order is stable shrinks, collapsing eventually to a line of tricritical points that separates the surfaces of first- and second-order phase transitions in the T-U-Δ space.
In Chapter 4 of the thesis the t-t’-U-Δ IHM on a square lattice is studied using DMFT+CTQMC. It is shown here that the presence of t’ frustrates the AF order for weak to intermediate correlations, resulting in the stabilization of the correlation induced PM phase over a broad region of the parameter space, although the AFI is always the stable phase for large enough U. In the intermediate coupling regimes, two other interesting phases show up as a result of the competition among U, Δ and t’ : a Ferrimagnetic metallic (FM) phase which has non-zero values of the uniform as well as staggered magnetization; and even more interesting, an…
In this thesis, we tackle various problems in strongly correlated electron systems, which can be addressed properly via the non-perturbative dynamical mean field theory (DMFT) approach using the continuous time quantum Monte Carlo method as an impurity solver. First, we revisit the old Nagaoka ferromagnetism problem in the U=oo Hubbard model and study the stability of the ferromagnetic state as a function of the temperature, the doping level, and the next-nearest-neighbor hopping t'. We then address the nature of the Mott transition in the two-dimensional Hubbard model at half-filling using cluster DMFT. Cluster DMFT can incorporate the short-range correlations beyond DMFT by extending the spatial range in which correlations are treated exactly to a finite cluster size. The non-local correlations reduce substantially the critical interaction U and modify the shape of the transition lines in the phase diagram. We then concentrate on the calculation of two-particle response functions from the ab initio perspective by means of computing the one-particle excitation spectrum using the combination of the density functional theory (DFT) and DMFT and extracting the two-particle irreducible vertex function from a local two-particle Green's function computed within DMFT. In particular, we derive the equations for calculating the magnetic/charge susceptibility and the pairing susceptibility in superconductivity. This approach is applied to the Hubbard model and the periodic Anderson model and we determine the phase diagram of magnetism and superconductivity in these models. We show that the superconducting phase is indeed stable near the magnetic phase where the pairing interaction mediated by spin fluctuations is dominantly enhanced. The non-local correlation effect to superconductivity is also discussed using the dual fermion approach and the dynamical vertex approximation. We finally apply the vertex function approach within DFT+DMFT to a Fe-based superconductor, BaFe2As2, and compute the dynamical magnetic susceptibility in this material. Our calculation results show a good agreement with the magnetic excitation spectra observed in a neutron scattering experiment. The response function calculation method derived in this thesis can capture both a localized and an itinerant nature of collective excitations in strongly correlated electron systems.
The main focus of this thesis is the detailed investigation of computational methods to tackle strongly correlated materials in which a rich variety of exotic phenomena are found. A many-body problem with sizable electronic correlations can no longer be explained by independent-particle approximations such as density functional theory (DFT) or tight-binding approaches. The influence of an electron to the others is too strong for each electron to be treated as an independent quasiparticle and consequently those standard band-structure methods fail even at a qualitative level. One of the most powerful approaches for strong correlations is the dynamical mean-field theory (DMFT), which has enlightened the understanding of the Mott transition based on the Hubbard model. For realistic applications, the dynamical mean-field theory is combined with various independent-particles approaches. The most widely used one is the DMFT combined with the DFT in the local density approximation (LDA), so-called LDA+DMFT. In this approach, the electrons in the weakly correlated orbitals are calculated by LDA while others in the strongly correlated orbitals are treated by DMFT. Recently, the method combining DMFT with Hedin's GW approximation was also developed, in which the momentum-dependent self-energy is also added. In this thesis, we discuss the application of those methodologies based on DMFT. First, we apply the dynamical mean-field theory to solve the 3-dimensional Hubbard model in Chap. 3. In this application, we model the interface between the thermodynamically coexisting metal and Mott insulator. We show how to model the required slab geometry and extract the electronic spectra. We construct an effective Landau free energy and compute the variation of its parameters across the phase diagram. Finally, using a linear mixture of the density and double-occupancy, we identify a natural Ising order parameter which unifies the treatment of the bandwidth and filling controlled Mott transitions. Secondly, we study the double-counting problem, a subtle issue that arises in LDA+DMFT. We propose a highly precise double-counting functional, in which the intersection of LDA and DMFT is calculated exactly, and implement a parameter-free version of the LDA+DMFT that is tested on one of the simplest strongly correlated systems, the H2 molecule. We show that the exact double-counting treatment along with a good DMFT projector leads to very accurate and total energy and excitation spectrum of H2 molecule. Finally, we implement various versions of GW+DMFT, in its fully self-consistent way, one shot GW approximation, and quasiparticle self-consistent scheme, and studied how well these combined methods perform on H2 molecule as compared to more established methods such as LDA+DMFT. We found that most flavors of GW+DMFT break down in strongly correlated regime due to causality violation. Among GW+DMFT methods, only the self-consistent quasiparticle GW+DMFT with static double-counting, and a new method with causal double-counting, correctly recover the atomic limit at large H-atom separation. While some flavors of GW+DMFT improve the single-electron spectra of LDA+DMFT, the total energy is best predicted by LDA+DMFT, for which the exact double-counting is known, and is static.
Correlated electron systems exhibit a variety of intriguing emergent phenomena, including Mott physics, pseudogap behavior, and unconventional superconductivity. One of the most widely used numerical approaches for investigating these systems is dynamical mean-field theory (DMFT). This dissertation addresses two related problems: how to develop better quantum impurity solvers for DMFT and how to use DMFT results to gain a better understanding of correlated electron systems. The first part of this dissertation develops numerical methods for solving quantum impurity problems, which constitute the central computational task in DMFT calculations. In Chap. 4, we develop a method that combines a tensor-train numerical integration scheme with the inchworm hybridization expansion for quantum impurity problems. Benchmarks on spinless, single-orbital, and multi-orbital impurity models demonstrate the convergence of the Green’s functions to exact or reference results and identify the regimes in which tensor-train compression of Feynman diagrams and the inchworm expansion itself are effective. In Chap. 5, we introduce the concept of counterterms within the hybridization expansion and apply it to a bare hybridization expansion solver that relies on a sum-of-exponentials integration technique. Tests on spinless, multi-orbital, and realistic embedding impurity problems show that counterterms can significantly reduce the required perturbative order and improve the efficiency of hybridization-expansion calculations. The second part of this dissertation applies the cluster extension of DMFT to the single-band Hubbard model. In Chap. 6, we introduce a fluctuation decomposition for the self-energy that eliminates the Fierz ambiguity encountered in conventional fluctuation diagnostics of the Hubbard model. The resulting analysis shows that spin fluctuations make an important contribution to the pseudogap self-energy, while multi-boson contributions are also necessary for a complete description. In Chap. 7, we analyze the spin-fermion coupling and its role in d-wave pairing. Our calculations show that, for antinodal fermions in the pseudogap regime, their coupling to antiferromagnetic spin fluctuations is strongly enhanced relative to the non-interacting limit. A modified Bethe–Salpeter equation analysis further shows that this enhanced spin–fermion coupling boosts the spin-fluctuation-induced contribution to the d-wave pairing susceptibility.