In this thesis, different wave-function based numerical methods are used to study many-body systems with bosonic degrees of freedom in 1d. The main focus is on the Holstein model of spinless fermions, where ground-state phases, nonequilibrium dynamics, and thermalization are investigated. Using matrix-product-state methods for the investigation of such models with electron-phonon coupling poses special numerical challenges. Local basis optimization, projected purification, and subspace expansion are used to overcome these challenges.
In this thesis, different wave-function based numerical methods are used to study many-body systems with bosonic degrees of freedom in 1d. The main focus is on the Holstein model of spinless fermions, where ground-state phases, nonequilibrium dynamics, and thermalization are investigated. Using matrix-product-state methods for the investigation of such models with electron-phonon coupling poses special numerical challenges. Local basis optimization, projected purification, and subspace expansion are used to overcome these challenges.
In this thesis, I investigate the transport and dynamical properties of electron-phonon coupled systems using numerical methods. I focus on the one-dimensional Holstein model, which incorporates both polaron and charge density wave (CDW) physics, and use matrix-product-state-based algorithms. In the first part, I present work where we combine purification with local basis optimization to study the finite-temperature properties of the model. This gives access to polaron spectral functions, polaron and bipolaron optical conductivity, and the energy-transport coefficient at finite filling. The second part focuses on the Holstein model coupled to two tight-binding leads. There, we investigate how current is transported through the structure and how CDW states break down. In the last part, we analyze the spread of an electron in a lattice and the decay of CDWs. We aim to understand if a full quantum mechanical treatment of the phonons is necessary or if trajectory-based algorithms are applicable.
In this dissertation a newly developed numerical method is presented, which is optimized for the time evolution of one-dimensional lattice systems with large local Hilbert spaces. This method extends the time-evolving block-decimation (TEBD) to include a local basis optimization (LBO), which has already been successfully combined with ground state methods. The algorithm is based on matrix product states (MPS), which can represent the quantum state of a one-dimensional chain in most cases with a number of parameters, that is not exponentially increasing with the chain length. The LBO causes a reduction of the simulation times that is linear in the bond dimension of the MPS. To demonstrate the advantages of this method, we apply the TEBD-LBO to electron-phonon (e-p) systems. In this thesis these are described by the Holstein model, which goes beyond semi-classical approximations and covers the full quantum statistics of the phonons. The understanding of the nonequilibrium dynamics of charge carriers coupled to lattice vibrations is of great importance for research areas like transport through quasi one-dimensional conductors, photo-generated phase transitions and time-resolved spectroscopy. First, the energy transfer from a highly excited electron to the phononic degrees of freedom on a small chain is studied. In the various parameter regimes different types of relaxation occur. In any case, after a certain time the system reaches a state, where on average no energy is exchanged between the electron and phonons. This can either mean a constant kinetic energy or oscillations with a constant amplitude and frequency. Next, long perfectly conducting leads without coupling to phonons were attached on both sides of the small chain. In the left lead an electron with density distribution in the shape of a Gaussian wave packet is injected with momentum towards the e-p coupled structure in the middle. This structure acts as an impurity in an otherwise perfectly conducting chain. The investigation shows resonance effects in the transmission and reflection at this impurity. Further, the electron can transfer a part of its energy permanently to the phonons, which results in a reduction of the velocity. Finally, two mechanisms are presented that lead to self-trapping of the electron on the e-p coupled structure.
The nonequilibrium effects of dissipation and drive play a key role in an immense variety of nanoscale and mesoscale quantum systems. To understand the behaviour of open quantum systems, we need accurate methods that capture the influence of the environment on the system, while managing the exponentially large Hilbert space required to describe the system. Tensor network algorithms offer an efficient way to approach this challenge. In this thesis, we develop and apply tensor network techniques to study the dynamics and steady states of various open quantum systems. The first part of the thesis focuses on the driven dissipative many body physics in coupled cavity arrays described by Born-Markov master equations. We extend transfer matrix product operator methods to Liouvillian dynamics, and utilize them to compute dynamical correlation functions and fluorescence spectrum of an infinite coupled cavity array in 1D. We also investigate thermalization, and observe the emergence of a quasi-thermal steady state with a negative effective temperature. In another study, we use infinite projected entangled pair state (iPEPS) methods to compute steady states of coupled cavity lattices in 2D. We find that a straightforward adaptation of iPEPS to Liouvillian dynamics is unstable, contradicting a recent publication in the field. The second part investigates more general systems involving strong couplings and structured environments that induce non-Markovian dynamics. We develop a powerful time-evolving matrix product operator (TEMPO) algorithm that builds on Feynman-Vernon influence functional formalism, and uses matrix product states (MPS) to represent the temporal non-Markovian correlations efficiently. We apply TEMPO to study the localization phase transition of the spin-boson model and the dynamics of two spatially separated two-level systems coupled to a common environment. Finally, we propose the Toblerone TEMPO algorithm, which extends TEMPO to many-body systems interacting with general bosonic environments.
Exact solutions of many-body problems hold a very special and significant place in condensed matter physics, helping us to understand the behaviors of interacting particles. In this thesis, we focus our attention on study the exact solutions of two interesting many-body problems in stronglycorrelated \n \nsystems. One is to calculate the exact correlation functions of \n \n1D chains of spinless fermions and hardcore bosons. The other is to calculate the single-particle dispersion relation of Hubbard model of spinless fermions on 2D square lattices.
Quantum many-body physics has been studied for many decades. A lot of intriguing phenomena have been observed and theories have been developed. Yet many problems remain unsolved, which is largely due to a lack of general efficient classical computation method. One-dimensional (1D) systems have also drawn much attention over the past few decades. One reason is that many unique strongly correlated quantum phenomena only appear in low dimensions. Another reason is that many exact results can be obtained in 1D for cross benchmarking, such as Bethe Ansatz, Bosonization, Bose-Fermi mapping, etc. In addition, many numerical methods, such as Matrix-Product-State based and Monte Carlo methods, work most efficiently in 1D. In recent years, 1D quantum gases have been realized in many cold atom labs, providing experimental motivation for their studies. One of the most mysterious assumptions about a system of identical quantum particles is that the wavefunction must be symmetric (for bosons) or anti-symmetric (for fermions). This means that no two fermions can occupy the same state, while multiple occupancy is allowed for bosons. Bosons and fermions are therefore generally very different. But in 1D this distinction could become rather subtle, provided that the bosonic multiple occupancy is suppressed, which can happen if strong repulsion exists between them. However, when considering spin degrees of freedom, things become more complicated. In this work, we develop a generalized Bose-Fermi mapping theory, under which, the charge degrees of freedom is mapped to a spinless fermions, while the spin degrees of freedom to an effective spin chain. This mapping works for arbitrary spin, arbitrary trapping potentials, arbitrary spin-dependent interaction potential, and for either spinor bosons or fermions. In the strong interaction limit, the wavefunction of the system is represented by the strong coupling ansatz wavefunction (SCAW), which takes the form of a direct product of spinless fermions wavefunction and spin wavefunction. Using this mapping technique, we study the dynamics and collective modes of the system. Furthermore, we develop a very efficient method to calculate the one-body density matrix, from which we can calculate the momentum distribution of the system.
In this thesis, strongly correlated quantum many-body systems in equilibrium and in out-of-equilibrium situations are investigated. This is done by applying and developing well established numerical methods. The focus of the thesis lies in the development and application of the density matrix renormalization group method (DMRG) to quantum many-body systems out of equilbrium. In this thesis, in addition to the DMRG, methods for the exact diagonalization of the Hamiltonian of the system, like the Lanczos- or the Jacobi-Davidson method, are treated. An extension of the Lanczos method makes it possible to treat the time evolution of strongly correlated quantum systems with an accuracy comparable to machine precision. This method is the basis for a possible extension of the DMRG for the treatment of systems out of equilibrium, the so-called adaptive time-dependent DMRG ("adaptive t-DMRG"). A second variant of the adaptive t-DMRG uses the Suzuki-Trotter decomposition of the time-evolution operator. An error analysis demonstrates that both methods, for a suitable choice of control parameters, have errors < 1% at the end of the time evolution in complicated quantities like, e.g., the momentum distribution function. We apply these numerical methods to investigate the quantum critical behavior of a variant of the Hubbard model and to treat two non-equilibrium situations of current interest. Extensive use of the DMRG makes it possible to clarify the quantum-critical behavior of the ionic Hubbard model; in particular, the numerical results demonstrate quite clearly that the correct scenario has two critical points; at the first critical point, only the charge degrees of freedom are critical, while at the second one only the spin degrees of freedom are critical. Quite surprisingly, the Mott-insulator phase in the strong-coupling regime shows a divergent electrical susceptibility. Next, the dynamics of a system of so-called soft-core bosons is treated. The particles initially are trapped in a deep box-potential and the dynamics is investigated after releasing them from this trapping potential. Similar to exact results obtained for so-called hard-core bosons, (quasi-)coherent matter waves emerge, demonstrating the possibility of realizing an atom laser in experiments on optical lattices. We find that the wave vector of the emerging matter wave can be tuned by changing the strength of the interaction between the particles. The second non-equilibrium situation is a so-called "quantum-quench" of a system of strongly correlated fermions, i.e., the evolution of the system after suddenly changing an intrinsic parameter like, e.g., the interaction strength between the particles, is investigated. In this case, we are mainly interested in the long-time behavior; in particular, the question arises whether, relying on Boltzmann's ergodic hypothesis, relaxation to a thermal state is obtained. We find that, in general, non-thermal final states are reached, which can be described by generalized Gibbs-Boltzmann ensembles.