This PhD thesis conducts a focused study of strongly correlated materials with localized electron orbitals. We have studied two real materials (LuNiO$_3$ and VO$_2$) and one model system, i.e., the Anderson impurity model. The thesis is divided into two parts. In the first part, we use the DFT+$U$ method, to calculate the equilibrium phase transitions of LuNiO$_3$ as an example of rare-earth nickelates under a substrate-induced strain, and the nonequilibrium phase transitions of VO$_2$ as an example of narrow-gap Mott insulators under laser-pulse-induced photoexcitations. The effect of adding $U$ is manifested in both materials as the change of band structure in response to the change of orbital occupancies of electrons, i.e., the soft band effect. This effect brings about competitions of electrons between different orbitals and gives rise to multiple metastable states. In the second part, we go beyond band theory and study how we can use the density matrix renormalization group (DMRG) method based on matrix product states (MPS) to perform real-time evolutions of the Anderson impurity model, towards the goal of building a real-time impurity solver for the nonequilibrium dynamical mean-field theory (DMFT). We study both the quenched and periodically driven single-impurity Anderson models (SIAM) and have obtained some complexity results based on a non-standard implementation of DMRG (the 4-MPS method) we developed in the thesis. We find in the star geometry of the SIAM that the ordering of the bath orbitals in the MPS affects dramatically the increase of entanglement entropy and thus the computational complexity of the simulation.
This PhD thesis conducts a focused study of strongly correlated materials with localized electron orbitals. We have studied two real materials (LuNiO3 and VO2) and one model system, i.e., the Anderson impurity model. The thesis is divided into two parts. In the first part, we use the DFT+U method, to calculate the equilibrium phase transitions of LuNiO3 as an example of rare-earth nickelates under a substrate-induced strain, and the nonequilibrium phase transitions of VO2 as an example of narrow-gap Mott insulators under laser-pulse-induced photoexcitations. The effect of adding U is manifested in both materials as the change of band structure in response to the change of orbital occupancies of electrons, i.e., the soft band effect. This effect brings about competitions of electrons between different orbitals and gives rise to multiple metastable states. In the second part, we go beyond band theory and study how we can use the density matrix renormalization group (DMRG) method based on matrix product states (MPS) to perform real-time evolutions of the Anderson impurity model, towards the goal of building a real-time impurity solver for the nonequilibrium dynamical mean-field theory (DMFT). We study both the quenched and periodically driven single-impurity Anderson models (SIAM) and have obtained some complexity results based on a non-standard implementation of DMRG (the 4-MPS method) we developed in the thesis. We find in the star geometry of the SIAM that the ordering of the bath orbitals in the MPS affects dramatically the increase of entanglement entropy and thus the computational complexity of the simulation.
For a long time strong electronic correlations in metals have mainly been associated with Mottness, the proximity to a Mott metal-insulator transition (MIT), where large Coulomb interactions induce the localization of charges. However, triggered by the discovery of the iron-based superconductors about ten years ago, it was realized that multi-orbital materials with only moderate Coulomb but sizeable Hund’s rule interactions – so-called Hund metals – allow for a distinct screening mechanism towards strong correlations: Hundness. Here, Hund’s rule constrains the spin rather than the charge dynamics. This discovery led to a vividly debated fundamental issue in the field of strongly correlated condensed matter systems, which is the main topic of the present thesis: what is the origin of strong correlations in the normal phase of Hund metals, Mottness or Hundness? And what are their decisive fingerprints? The goal of this dissertation is twofold. First, we present and advance our method: the numerical renormalization group (NRG) as viable real-frequency multi-band impurity solver for dynamical mean-field theory (DMFT), a common approach to tackle strongly correlated systems. Second, we apply DMFT+NRG to shed light on the Hund-metal problem raised above. In the first part of this thesis we present our state-of-the-art NRG solver, which offers direct access to data with unprecedented real-frequency spectral resolution at arbitrarily low energies and temperatures in contrast to commonly used Quantum Monte Carlo solvers. It is based on matrix product states and exploits non-abelian symmetries to reduce numerical costs. In the case of orbital symmetry, this allows us to treat multi-band models with more than two bands, and thus to tackle the Hund-metal problem for the first time with NRG. For multi-band models without orbital symmetry, an “interleaved” scheme of NRG (iNRG) was recently developed, dramatically increasing the numerical efficiency. Remarkably, the accuracy of iNRG is comparable to standard NRG, as we reveal in a detailed study. This finding establishes iNRG as a promising DMFT solver for material-specific model simulations. In the second part of this thesis we study a minimal toy model for Hund metals with DMFT+NRG, the orbital-symmetric three-band Hubbard-Hund model (3HHM) close to a lattice filling of 1/3. Our major insight is “spin-orbital separation” (SOS), a Hund’s-ruleinduced two-stage Kondo-type screening process, in which orbital screening occurs at much higher energies than spin screening. In Hund metals, i.e. far from a MIT phase boundary, SOS thus causes large electron masses by strongly reducing the coherence scale below which a Fermi liquid is formed. Further, it opens up a broad incoherent and strongly particle-hole asymmetric intermediate energy regime that reaches up to bare excitation scales. This SOS regime shows fractional power-law behavior and is characterized by resilient “Hund quasiparticles” with itinerant orbital degrees of freedom coupled non-trivially to quasi-free large spins. At zero temperature, the local density of states exhibits a two-tier quasiparticle peak on top of a broad incoherent background. In contrast, in Mott-correlated metals, i.e. close to the MIT phase boundary, the SOS regime becomes negligibly small and the Hubbard bands are well separated. These findings lead to distinct signatures of Hundness and Mottness in the temperature dependence of ARPES spectra, static local susceptibilities, resistivity, thermopower and entropy, many of which were also found in realistic simulations of the archetypal Hund- and Mott-correlated materials, Sr2RuO4 and V2O3. In summary, we provide evidence that and elucidate how Hundness evokes strong correlation effects in Hund metals. This might help to better interpret experimental results and guide superconducting theories.
Numerical studies on strongly correlated fermionic systems are very complicated and still provide essential problems. The main reason is the exponential growth of the un- derlying Hilbert state space with the system size and the fermionic sign problem for Monte Carlo studies. Among the most widely employed numerical techniques for study- ing two-dimensional quantum many-body systems are cluster extensions of the dynamical mean-field theory (DMFT), e.g. dynamical cluster approximation (DCA). They map an infinitely large multi-dimensional lattice problem to a one-dimensional impurity problem. In 2015 it was shown that the density matrix renormalisation group (DMRG) used as an impurity solver for DMFT (DMFT+DMRG) on the imaginary-frequency axis allows to solve multi-site and multi-band problems extremely fast compared to other solvers. Within this thesis, we further develop this DMRG+DMFT approach to apply the method on real material settings. The step from artificial, completely degenerate multi-band mod- els with simple dispersion relations on a Bethe lattice, studied in 2015, to systems with realistic band structures and lifted degeneracies involves more challenges than originally suspected. In this thesis, we will first recapitulate relevant methods for our approach like matrix prod- uct states, the density matrix renormalisation group and several time evolution methods. In this context we will present several improvements ranging from optimised time evo- lutions to entanglement based optimisations of tensor networks. Second, we will present a very detailed description of the dynamical mean field theory. We will focus on both methodological aspects and implementation details. This chapter is intended to allow other researcher to implement their own DMFT code using DMRG as an impurity solver. Third, we will discuss three different models to show the extent of problems DMRG+ DMFT is able to solve. We will focus on multi-site DCA calculations in the case of the two-dimensional Hubbard model and show that DMRG allows to tackle systems with intermediate interaction strengths at low temperatures, which are unsolvable with other solvers. In the second case, the real material Sr2VO4, we will show the first two-site DCA results for a realistic three-band model. In contrast to assumptions, partly reintroducing the momentum dependence of the self-energy does not improve agreement between exper- imental observations and theoretical results. Finally, we will move on to another realistic three-band model, which describes Sr2RuO4, to show how to deal with the influence of spin-orbit coupling on DMFT. We will present the first low-temperature results for this material and will confirm previous results of simplified model calculations.
Dynamical mean-field theory (DMFT) is a widely used, nonperturbative computational framework for strongly correlated electron systems. This thesis extends DMFT in two directions: treating disorder and higher-order correlators away from equilibrium, and using near-term quantum computers as impurity solvers. In the first part, we develop two methodological advances for the nonequilibrium DMFT treatment of disordered, interacting systems. The first overcomes the central computational bottleneck of the recently developed nonequilibrium DMFT plus coherent potential approximation (DMFT+CPA) by replacing its explicit numerical disorder averaging with an exact analytical integration, exploiting a normalization property of the Schwinger-Keldysh contour; applied to a disorder-and-interaction quench of the Anderson-Hubbard model, this reveals that weak random disorder promotes thermalization. The second demonstrates that the out-of-time-order correlator (OTOC) can be successfully computed within the existing DMFT+CPA framework, by generalizing the Schwinger-Keldysh contour to a double-folded structure; as a proof-of-principle demonstration, weak disorder is found to accelerate the OTOC's decay in the half-filled Anderson-Hubbard model in the intermediate-coupling regime. In the second part, we explore the use of near-term quantum computers as DMFT impurity solvers. A stable real-time DMFT iteration scheme, designed for compatibility with quantum hardware, is first developed and validated classically against the metal-to-insulator transition. A symmetry-adapted variational quantum eigensolver (VQE) is then benchmarked as the Anderson impurity solver for the converged DMFT Hamiltonian, and is found to reproduce ground-state energies accurately and to extract the single-particle Green's function reliably at intermediate to strong coupling, while accurate ground-state energies are shown not to guarantee accurate dynamics at weak coupling, a key challenge for extending such approaches to correlated metals. Together, we demonstrate that DMFT can be methodologically extended to study disordered and interacting systems away from equilibrium, and to incorporate near-term quantum hardware as a genuinely new computational resource to enhance its capabilities.
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.
Strong electronic correlations are the root of many fascinating phenomena of condensed matter physics. However, they also make calculations notoriously difficult. This thesis is concerned with the development and improvement of computational methods for strongly correlated electron systems. After a brief recapitulation of the Green’s function formalism,which is the theoretical basis of most such methods, we discuss the analytical continuation from imaginary to real frequencies. Especially the maximum entropy method is explained in detail and an implementation there of is presented.The remainder of this thesis deals with methods to compute Green’s functions on thei maginary frequency axis. First, the Anderson impurity model (AIM) is considered. It can be solved numerically – a task for which several methods exist. We focus on continuous-time quantum Monte Carlo with worm sampling, through which various kinds of correlation functions can be calculated. On this basis we derive and implement symmetric improved estimators that allow for the computation of the self-energy and vertex functions where the level of stochastic noise is largely reduced with respect to conventional estimators.As a next step the dynamical mean-field theory (DMFT) is introduced as a method to capture local correlations in the Hubbard model (HM) by self-consistently mapping it onto the AIM. We study the convergence behavior of this iterative procedure and the improvement of convergence due to mixing. Then, we compare electron scattering rates from DMFT in the weak-coupling regime of the square-lattice HM to results from Boltzmann scattering theory.While local correlations account for many interesting phenomena, such as the Mott-Hubbard metal-insulator transition, they are not sufficient to correctly describe, e.g., the physics of phase transitions in two- or three-dimensional systems. It is thus the final step in this thesis to also include nonlocal correlations by means of the dynamical vertex approximation (DΓA). For this we develop a new self-consistency scheme, which is then applied to several systems: We study the square-lattice HM, where we compare to benchmark methods in the weak-coupling regime, but also present results for intermediate and strong coupling. Furthermore, a two-band model with doping and a three-band Wannier model for strontium vanadate are studied. As an application of our new self-consistent DΓA, we study the effects of geometrical frustration in the kagome lattice, where we also compare our results to determinant quantum Monte Carlo.
Strongly correlated electron systems host a plethora of fascinating physical phenomena and pose formidable challenges in their theoretical analysis. The challenges originate from the inherent complexity of the quantum many-body problem - no classical computer will ever be able to fully simulate these systems - and the lack of an effective single-particle picture, as the strong mutual interactions make it impossible to regard the electrons as independent from each other. As a consequence, most systems of correlated electrons can only be tackled approximately and numerically. In this thesis, we develop a set of numerical methods for strongly correlated electrons, which are inspired by the renormalization group (RG) idea of including degrees of freedom successively from high to low energies. This enables an efficient organization of the diverse fluctuations and is key for an accurate treatment of interacting quantum systems, where collective behavior and composite objects emerge at energy scales far below those of the microscopic constituents. In a first part, we consider the functional renormalization group (fRG), a versatile framework to study the flow of correlation functions upon modulating the underlying action. Though widely used, it has often acted more as a qualitative rather than quantitative method, due a nontransparent approximation induced by truncating the hierarchy of flow equations. We develop an iterative multiloop fRG (mfRG) scheme, which ameliorates this approximation and eliminates many of the drawbacks of fRG experienced hitherto. In particular, it restores the independence of results on the choice of RG regulator and establishes a rigorous relation to the parquet formalism. Furthermore, we show how to derive the flow equations directly from self-consistent many-body relations. This establishes a form of diagrammatic resummations at the two-particle level which circumvents ill-behaved two-particle-irreducible vertices. An application to the prototypical two-dimensional Hubbard model illustrates how our multiloop scheme elevates the fRG approach to correlated electron systems to a quantitative level. Secondly, we employ the numerical renormalization group (NRG), based on the iterative diagonalization of impurity Hamiltonians, in conjunction with the dynamical mean-field theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, real-frequency impurity solver for DMFT. It has been pivotal to the understanding of Hund metals, where strong correlations arise from Hund's rules even at moderate Coulomb repulsion. Building on recent methodological advances, we extend the range of application of DMFT+NRG from orbital-degenerate models to more realistic setups: We first study orbital differentiation in a three-orbital Hund-metal model and unravel key effects of the orbital-selective Mott transition. In a real-materials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hund-metal material Sr2RuO4. We particularly follow its RG flow to the Fermi-liquid regime at previously inaccessible low temperatures and generally present DFT+DMFT+NRG as a new computational paradigm for strongly correlated materials. As a side project of our fRG work, we develop an algorithm to count Feynman diagrams from closed many-body relations, which reveals the surprising outcome that totally irreducible contributions are responsible for the factorial growth in the number of diagrams. Additionally, we use NRG to study transport through three-level quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium.