The emergence of order in materials with strongly-correlated electrons in out-of-equilibrium situations inspires a lot of new research, both experimental and theoretical. The main goal of this theoretical research project is to better understand related questions in the dynamics in a strongly correlated many-body state after a photoexcitation has occured. Such situations are usually not fully explainable in a mean-field picture nor analytically solvable. Hence, advanced numerical techniques are necessary. In this work, we investigate non-equilib- rium situations after photoexcitations. In order to model a hypothetical one-dimensional (1D) manganite, we chose the 1D Hubbard model with nearest-neighbor interaction and a staggered magnetic field. The photoexcitation is modeled in two different ways: First we investigate sud- den, local excitations, and afterwards we study a semi-classical approach by applying the Peierls substitution, which leads to a time-dependent Hamiltonian. All simulations are performed with an implementation of the time-dependent density-matrix renormalization group (DMRG), which is formulated by tensor-network states (TNSs), namely matrix-product states (MPSs) and matrix-product operators (MPOs). The framework used offers the possibility that every MPO can be externally described by finite-state machines (FSMs), hence it is extremely flexible. In this thesis, we explain how to perform exact FSM arithmetics, and how to compress the resulting FSMs. Based on MPSs and FSMs, a quantum-computer simulator (QCS) is introduced, which is mainly used as a universal tool for (MPS)-quantum- state manipulations. From the investigations with the sudden, local excitations, we learned that the electron-electron interaction is responsible for a rapid relaxation of the magnetic moment of the individual bands. Nevertheless, this relaxation can be stalled via a stronger magnetic microstructure. By applying a spin-selective photoexcitation via the Peierls substitution, we are able to induce a meta-stable charge-density wave (CDW) pattern if a magnetic microstructure is present. For a small, but finite interaction, we find a decay channel for the doublon-based part of the CDW, which still leaves a finite pattern. For large interaction, nearly no doublons are created by the photoexcitation. In the opposite limit, i.e., the non-interacting case, the two spin species are decoupled. Hence, in both limits the decay channel does not weaken the CDW and we find the pattern to be stable up to the times we can treat.
The emergence of order in materials with strongly-correlated electrons in out-of-equilibrium situations inspires a lot of new research, both experimental and theoretical. The main goal of this theoretical research project is to better understand related questions in the dynamics in a strongly correlated many-body state after a photoexcitation has occured. Such situations are usually not fully explainable in a mean-field picture nor analytically solvable. Hence, advanced numerical techniques are necessary. In this work, we investigate non-equilib- rium situations after photoexcitations. In order to model a hypothetical one-dimensional (1D) manganite, we chose the 1D Hubbard model with nearest-neighbor interaction and a staggered magnetic field. The photoexcitation is modeled in two different ways: First we investigate sud- den, local excitations, and afterwards we study a semi-classical approach by applying the Peierls substitution, which leads to a time-dependent Hamiltonian. All simulations are performed with an implementation of the time-dependent density-matrix renormalization group (DMRG), which is formulated by tensor-network states (TNSs), namely matrix-product states (MPSs) and matrix-product operators (MPOs). The framework used offers the possibility that every MPO can be externally described by finite-state machines (FSMs), hence it is extremely flexible. In this thesis, we explain how to perform exact FSM arithmetics, and how to compress the resulting FSMs. Based on MPSs and FSMs, a quantum-computer simulator (QCS) is introduced, which is mainly used as a universal tool for (MPS)-quantum- state manipulations. From the investigations with the sudden, local excitations, we learned that the electron-electron interaction is responsible for a rapid relaxation of the magnetic moment of the individual bands. Nevertheless, this relaxation can be stalled via a stronger magnetic microstructure. By applying a spin-selective photoexcitation via the Peierls substitution, we are able to induce a meta-stable charge-density wave (CDW) pattern if a magnetic microstructure is present. For a small, but finite interaction, we find a decay channel for the doublon-based part of the CDW, which still leaves a finite pattern. For large interaction, nearly no doublons are created by the photoexcitation. In the opposite limit, i.e., the non-interacting case, the two spin species are decoupled. Hence, in both limits the decay channel does not weaken the CDW and we find the pattern to be stable up to the times we can treat.
We investigate the behavior of 1D correlated systems following a photo-excitation by combining ab-initio methods, time-dependent matrix product state (MPS) approaches, analytical insights from quantum Boltzmann equations, and molecular dynamics (MD) simulations to describe the dynamics on different time scales ranging from femtoseconds up to nanoseconds. This is done for manganite systems in the material class Pr1−xCaxMnO3. We derive one-dimensional ab-initio model Hamiltonians for which we compute the ground states at different values of the doping using MD simulations. At half doping, we obtain a magnetic microstructure of alternating dimers which we use as a starting point to formulate a one-dimensional Hubbard-type model. In this strongly correlated 1D system we address the formation of quasiparticles after photoexcitations. We introduce a quasiclassical model for the absorption process of light which takes into account the backaction of the electrons to the electromagnetic field and compare the results with the ones obtained with a Peierls substitution ansatz in which we neglect the backaction. Taking this effect into account leads to a stronger response since the emitted radiation can be re-absorbed by neighboring particles. We use the resulting state as starting point for the short-time dynamics which we obtain from 'numerically exact' MPS calculations. The dynamics is analyzed concerning the formation and lifetime of quasi-particles which form a starting point for the description of the relaxation process via a linearized quantum Boltzmann equation. In this way, our work constitutes a first step to building a unifying theoretical framework for the description of photoexcitations in strongly correlated materials over a wide range of time scales, capable of making predictions for ongoing experiments investigating pump-probe situations and unconventional photovoltaics.
In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results.
We investigate the evolution of a photoexcitation in correlated materials over a wide range of time scales. The system studied is a one-dimensional model of a manganite with correlated electron, spin, orbital, and lattice degrees of freedom, which we relate to the three-dimensional material ${\mathrm{Pr}}_{1\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{MnO}}_{3}$. The ground-state phases for the entire composition range are determined and rationalized by a coarse-grained polaron model. At half doping a pattern of antiferromagnetically coupled Zener polarons is realized. Using time-dependent density-matrix renormalization group (tDMRG), we treat the electronic quantum dynamics following the excitation. The emergence of quasiparticles is addressed, and the relaxation of the nonequilibrium quasiparticle distribution is investigated via a linearized quantum-Boltzmann equation. Our approach shows that the magnetic microstructure caused by the Zener polarons leads to an increase of the relaxation times of the excitation.
In this thesis we numerically study non-equilibrium phenomena in one-dimensional quantum many-body systems. We discuss extensions of the concepts of spectral functions and of the optical conductivity to time-dependent non-equilibrium setups and compute these quantities by means of matrix product state (MPS)-based methods. In a first application we study the effect of a spin-selective particle-hole-like excitation on a Hubbard model with an additional magnetic superstructure. Following the excitation, this model exhibits a stable charge density, as well as a stable spin density pattern mirroring the superstructure. For interacting systems we find an additional in-gap signal in the spectral function in the spin direction opposite to the one we applied the excitation in, which also persists over time. Our results are confirmed by a computation of the corresponding optical conductivity. We interpret our results as a trace of an exciton, which we support by an analysis in the atomic limit, and therefore refer to our findings as a peculiar spinful exciton. In contrast to many other setups, no long-range Coulomb term is needed for this exciton to form. Second, we apply the same calculus to compute non-equilibrium spectral functions of a Floquet-driven system of interacting spinless fermions in the charge density wave (CDW) Mott insulator phase, where we model the driving through a Peierls substitution ansatz. We find good agreement of our results with computations from an effective model obtained from Floquet theory, as well as Floquet side bands. These, however, do not show at all times as expected. Furthermore, another in-gap signal occurs, that is somewhat reminiscent of free particles, which goes hand in hand with a melting of the CDW pattern.
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.
The first part of the thesis is dedicated to the study of Matrix Product State (MPS). Initially named "finitely correlated state", MPS is the basis of the Tensor Network State that is widely used to represent quantum many-body wave functions, and is also the essential concept in powerful numerical tools like Density-Matrix Renormalization Group. It is an ansatz that only samples a very small portion of the complete Hilbert space, but can represent ground states of 1-dimensional gapped local Hamiltonians efficiently. We will first look at the quantum fluctuation of observables in such states. In particular, we consider the Full Counting Statistics of MPS, which is separated into a bulk and boundary term. We identify a central limit theorem like behavior in the limit of large system sizes, and write the corrections of the central limit theorem for a finite size system in terms of the Edgeworth series. We also show that, for special cases of MPSs, like the famous Affleck, Lieb, Kennedy and Tasaki state, because of the topological nature of the state, this description is no longer valid. Next we look at the time-evolution of an MPS under time-dependent Hamiltonians. We show that in the "injective" case, the Schrodinger equation can be written in terms of the MPS matrices in an interesting way. We show, however, it can never produce an exact time-evolution that changes the entanglement structure of the state. The second part of the thesis focuses on X-ray absorption and scattering. We are particularly interested in the resonant inelastic X-ray scattering (RIXS), which has been rapidly developing recently. However, there is still an on-going debate about whether the RIXS spectrum of cuprate systems shows collective or quasi-particle physics. We first use a simple perturbation method to get a basic idea about the physics of the process. Then we use a determinant method and the quasi-particle picture to study the RIXS response of a variety of cuprate systems, like CLBLCO, YBCO, and Bi-2201, and find excellent agreement with experiments. We also develop a method to account for the superconducting gap within a meanfield approach and study it in detail for a model of p-wave superconductor. Although the gap is small compared to other band structure parameters, the effect of the gap is rather significant.