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the famous Affleck, Lieb, Kennedy and Tasaki state, because of the topological nature of the state, this description is no longer valid.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.
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.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.
The term Tensor Network States (TNS) refers to a number of families of states that represent different ansätze for the efficient description of the state of a quantum many-body system.Matrix Product States (MPS) are one particular case of TNS, and have become the most precise tool for the numerical study of one dimensional quantum many-body systems, as the basis of the Density Matrix Renormalization Group method.Lattice Gauge Theories (LGT), in their Hamiltonian version, offer a challenging scenario for these techniques.While the dimensions and sizes of the systems amenable to TNS studies are still far from those achievable by 4-dimensional LGT tools, Tensor Networks can be readily used for problems which more standard techniques, such as Markov chain Monte Carlo simulations, cannot easily tackle.Examples of such problems are the presence of a chemical potential or out-of-equilibrium dynamics.We have explored the performance of Matrix Product States in the case of the Schwinger model, as a widely used testbench for lattice techniques.Using finite-size, open boundary MPS, we are able to determine the low energy states of the model in a fully non-perturbative manner.The precision achieved by the method allows for accurate finite size and continuum limit extrapolations of the ground state energy, but also of the chiral condensate and the mass gaps, thus showing the feasibility of these techniques for gauge theory problems.
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.
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 research work in this thesis is based on strongly interacting quantum lattice systems. The biggest part of research was conducted using state-of-the-art tensor network simulations. Tensor networks provide efficient and highly accurate representations of quantum states when any simply connected patch of the quantum state is slightly entangled. Matrix Product States (MPS) is a tensor network representation for quantum states which is quasi-exact for one-dimensional systems when entanglement entropy of any bipartition of the state follows ``area-law". Projected Entangled Pair States (PEPS) is the extension of MPS for higher dimensional systems, where entanglement entropy area-law is non-trivial. These formalisms and their relevant ground state optimization techniques, Density Matrix Renormalization Group (DMRG) for MPS and Simple Update (SU) with Tensor Renormalization Group (TRG) for PEPS are thoroughly analysed. Entanglement entropy area-law is usually obeyed by the ground state of local Hamiltonians, while generic highly excited states follow a volume-law (they span a finite part of the Hilbert space). Recently, a class of interacting system was shown to undergo a dynamical phase transition (Many Body Localization) where the entropy of every eigenstate follows an area-law. This transition is achieved when the system is highly disordered and the quantum many body state becomes localized similarly to the free particle Anderson localization. The area-law property makes highly excited eigenstates of Many Body Localized (MBL) systems efficiently represented by the MPS ansatz. We develop a highly optimized algorithm (eDMRG) which goes beyond ground state optimization and successfully target eigenstates in any part of the spectrum. This algorithm is used together with analytical calculations, based on local integrability of MBL systems, to identify the universal behaviour of the entanglement spectrum of highly excited eigenstates in MBL systems. In the second part we study interacting bosons in 2D optical lattices. Strongly interacting bosons are simulated using the Bose-Hubbard (BH) model when the interactions are strictly local and the Extended Bose-Hubbard (EBH) model, when additional dipolar interactions are present. The ground states of BH/EBH Hamiltonians in a hexagonal lattice are studied. The phase spectrum includes various insulating and critical phases which are studied in detail using infinite-PEPS ansatz. Additional results on entanglement entropy scaling with respect to the filling of the lattice are presented. Finally, a strongly interacting Harper-Hofstadter Hamiltonian is realized by combining synthetic fields with the BH model. The homogeneity of the system is then broken by a parabolic trapping potential, similar to the ones used in cold atom experiments. Using time-dependent Gutzwiller ansatz (GA) the expansion dynamics of the cloud in large square lattices are studied. In contrast to the expansion dynamics of the BH model, it is found that the synthetic fields generate a self-trapping effect. Using phenomenology and simulations the dynamics are studied in the hard-core and soft-core limit.