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.
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.
The possibility to form excitons in photoilluminated correlated materials is central from fundamental and application oriented perspectives. In this paper we show how the interplay of electron-electron interactions and a magnetic superstructure leads to the formation of a peculiar spinful dark exciton, which can be detected in ARPES-type experiments and optical measurements. We study this by using matrix product states (MPS) to compute the time evolution of single-particle spectral functions and of the optical conductivity following an electron-hole excitation in a class of one-dimensional correlated band-insulators, simulated by Hubbard models with on-site interactions and alternating local magnetic fields. An excitation in only one specific spin direction leads to an additional band in the gap region of the spectral function only in the spin direction unaffected by the excitation and to an additional peak in the optical conductivity. As both is formed only after the excitation, this is interpreted as a dark exciton, which shows only in one spin direction. Recombination of the excitation happens on much longer time scales than the ones amenable to MPS. We discuss implications for experimental studies in correlated insulator systems.
Periodically driven quantum systems provide a versatile platform for engineering and exploring non-equilibrium states of matter that have no equilibrium counterparts– a field commonly re- ferred to as Floquet engineering. The research presented in this thesis addresses the microscopic interpretation of experimental spectroscopic signatures, the stability of these driven states in the presence of strong many-body interactions, and the emergence of novel forms of non-equilibrium criticality encoded in quantum entanglement. The first part of the thesis focuses on the interpretation of time- and angle-resolved photo- emission spectroscopy (tr-ARPES), the primary experimental tool for detecting Floquet–Bloch sidebands (FBs). The measured signal reflects interference between processes occurring both inside the material and in the vacuum. In particular, Floquet-dressed initial states coexist and interfere with Volkov-dressed final states arising from the laser-assisted photoelectric effect (LAPE). Their experimental identification requires careful comparison with theoretical predic- tions. Using graphene as a model Dirac material, we perform a systematic comparison between two theoretical frameworks: the first-order perturbative Born approximation (PB1) and time- dependent nonequilibrium Green’s functions (tdNEGF), and compare them with experimental results. We disentangle intrinsic Floquet states, LAPE, and their interference, and quantify how polarization, incidence angle, and near-surface screening shape the observed spectra. Our findings show that, once matrix elements are included, both methods capture qualitative trends, but tdNEGF is necessary to describe the full energy–momentum-resolved features, including hybridization gaps. While non-interacting systems exhibit well-defined FBs, interacting systems absorb energy and can evolve toward an infinite-temperature state. To study this competition between coherent driving and many-body heating, in the second part we consider a one-dimensional chain of interacting spinless fermions using exact diagonalization (ED) and matrix product state (MPS) methods. We compute time-dependent spectral functions and correlation measures, and compare the spectral width of the many-body excitation continuum with the driving frequency to define low- and high-frequency regimes. In our parameter regime, low-frequency driving leads to strong overlap between Floquet sectors, resulting in rapid heating and loss of coherence. For weakly populated continua, we find a transient timescale for the suppression of FBs. When the driving frequency exceeds the continuum width, FBs are well separated and long-lived. Thefinalpartofthethesisinvestigatesnon-equilibriumphenomenafromaninformation-theoretic perspective. Wereportanewclassofdynamicalphasetransitions,termedtemporalentanglement transitions (TET) characterized by non-analytic reorganizations of the entanglement spectrum, in a periodically driven transverse-field Ising chain. These transitions occur in the spectrum of the entanglement Hamiltonian and are signaled by dynamical spontaneous symmetry breaking, marked by the closure of the Schmidt gap and parity flips. These transitions remain invisible to conventional local observables. Finite-size scaling reveals that TET belong to a universality class with a correlation-length exponent ν = 1, mirroring equilibrium Ising universality despite their purely dynamical origin.
By utilizing time-dependent tensor-network algorithms in the infinite matrix-product-state representation, we theoretically investigate the pump-probe spectroscopy of the one-dimensional extended Hubbard model at half filling. Our focus lies on nonequilibrium optical conductivity and single-particle excitation spectra. In the spin-density-wave (SDW) phase, we identify an in-gap state in the nonequilibrium optical conductivity due to the formation of excitons (or doublon-holon pairs), generated by the pulse through nonlocal interactions. In the strong-coupling regime, we discern additional multiple in-gap and out-of-gap states. In the charge-density-wave (CDW) phase, we detect not only an in-gap state but also a finite Drude weight, which results from the dissolution of charge order by photoexcitation. Analyzing time-dependent single-particle excitation spectra directly in the thermodynamic limit confirms the origin of these new states in the SDW and CDW phases as the excitation of newly emerged dispersions. We illustrate that the pump-probe spectroscopy simulations in the thermodynamic limit furnish unambiguous spectral structures that allow for direct comparison with experimental results, and the integration of nonequilibrium optical conductivity and time- and angle-resolved photoemission spectroscopy provides comprehensive insights into nonequilibrium states.
Although charge-spin separation has important consequences for the properties of one-dimensional (1D) Mott-Hubbard insulators, matrix elements in some of its dynamical correlation functions involve the coupling of spin and charge degrees of freedom. The corresponding interplay of the 1D Mott-Hubbard insulator's charge and spin degrees of freedom is an issue of both fundamental and technological interest, for instance, concerning their dynamics at subpicosecond timescales. On the other hand, the up- and down-spin one-particle spectral functions are the simplest dynamical correlation functions that involve excitation of both the charge and spin degrees of freedom. They are thus suitable to extract basic useful physical information on the above interplay at finite magnetic fields. Here the line shape of such functions at and in the $(k,\ensuremath{\omega})$ plane's vicinity of their cusp singularities is studied for the Mott-Hubbard insulator described by the 1D Hubbard model with one fermion per site at zero and finite magnetic fields. At zero field, they can be accessed in terms of electrons by photoemission experiments. At finite field, such functions and corresponding interplay of correlations and magnetic-field effects refer to an involved nonperturbative many-particle problem that is poorly understood. The Mott-Hubbard gap that separates the addition and removal spectral functions is calculated for all spin densities and interaction values. The qualitative differences in the one-particle properties of the Mott-Hubbard insulator and corresponding doped insulator are also investigated. The relation of our theoretical results and predictions to both condensed-matter and ultracold spin-1/2 atom systems is discussed.
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.
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.