We consider several exactly solvable models of strongly correlated electrons in one dimension, such as the Heisenberg XXX model, the supersymmetric t-J model and the Hubbard model. These models can be solved by using the method of graded algebraic Bethe ansatz. We use it to design graded tensor networks which can be contracted approximately to obtain a Matrix Product State. This overcomes a major shortcoming of current density matrix renormalization group (DMRG) methods which work well on the ground states, but have difficulty working with the excited states of such models. In addition, observables such as correlation functions are important as they are experimentally measurable, but have been analytically described in the double scaling limit only. Moreover, these analytical results are mostly expressed in the form of determinants, which are numerically inefficient to compute. With the tensor network description of the spin models, we can efficiently compute any expectation value of the eigenstates on finite length lattices for direct comparison with laboratory results. As a proof of principle, we calculate correlation functions of ground states and excited states of such models on finite lattices of lengths in an intermediate regime which are of experimental interest. | 108 pages
The algebraic Bethe ansatz is a prosperous and well-established method for solving one-dimensional quantum models exactly. The solution of the complex eigenvalue problem is thereby reduced to the solution of a set of algebraic equations. Whereas the spectrum is usually obtained directly, the eigenstates are available only in terms of complex mathematical expressions. This makes it very hard, in general, to extract properties from the states, for example, correlation functions. In our work, we apply the tools of tensor-network states to describe the eigenstates approximately as matrix product states. From the matrix product state expression, we then obtain observables like the structure factor, dimer-dimer correlation functions, chiral correlation functions, and one-particle Green function directly.
This thesis treats the classical simulation of strongly-interacting many-body quantum-mechanical systems in more than one dimension using matrix product states and the more general tensor product states. Contrary to classical systems, quantum many-body systems possess an exponentially larger number of degrees of freedom, thereby significantly complicating their numerical treatment on a classical computer. For this thesis two different representations of quantum many-body states were employed. The first, the so-called matrix product states (MPS) form the basis for the extremely successful density matrix renormalization group (DMRG) algorithm. While originally conceived for one-dimensional systems, MPS are in principle capable of describing arbitrary quantum many-body states. Using concepts from quantum information theory it is possible to show that MPS provide a representation of one-dimensional quantum systems that scales polynomially in the number of particles, therefore allowing an efficient simulation of one-dimensional systems on a classical computer. One of the key results of this thesis is that MPS representations are indeed efficient enough to describe even large systems in two dimensions, thereby enabling the simulation of such systems using DMRG. As a demonstration of the power of the DMRG algorithm, it is applied to the Heisenberg antiferromagnet with spin
S=1/2 on the kagome lattice. This model's ground state has long been under debate, with proposals ranging from static spin configurations to so-called quantum spin liquids, states where quantum fluctuations destroy conventional order and give rise to exotic quantum orders. Using a fully
SU(2)-symmetric implementation allowed us to handle the exponential growth of entanglement and to perform a large-scale study of this system, finding the ground state for cylinders of up to 700 sites. Despite employing a one-dimensional algorithm for a two-dimensional system, we were able to compute the spin gap (i.e. the energy gap to the first spinful excitation) and study the ground state properties, such as the decay of correlation functions, the static spin structure factors, and the structure and distribution of the nearest-neighbor spin-spin correlations. Additionally, by applying a new tool from quantum information theory, the topological entanglement entropy, we could also with high confidence demonstrate the ground state of this model to be the elusive gapped
Z2 quantum spin liquid with topological order. To complement this study, we also considered the extension of MPS to higher dimensions, known as tensor product states (TPS). We implemented an optimization algorithm exploiting symmetries for this class of states and applied it to the bilinear-biquadratic-bicubic Heisenberg model with spin
S=3/2 on the
z=3 Bethe lattice. By carefully analyzing the simulation data we were able to determine the presence of both conventional and symmetry-protected topological order in this model, thereby demonstrating the analytically predicted existence of the Haldane phase in higher dimensions within an extended region of the phase diagram. Key properties of this symmetry-protected topological order include a doubling of the levels in the entanglement spectrum and the presence of edge spins, both of which were confirmed in our simulations. This finding simultaneously validated the applicability of the novel TPS algorithms to the search for exotic order.
The simulation and numerical study of large, strongly correlated quantum systems containing Fermions or using real-time evolution in finite dimensions is still an essentially unsolved problem, primarily due to the exponential growth of the Hilbert state space with system size and the occurrence of the so-called sign problem in Monte Carlo studies. In this area, the use of tensor-network methods, for one-dimensional systems chief among them the density matrix renormalisation group (DMRG) and matrix-product states (MPS), has grown in importance in recent years. This thesis first recapitulates the use of non-abelian symmetries such as SU(2)-Spin in arbitrary tensor networks with an extensive review of the published literature including detailed algorithms and implementation hints. Implementing such symmetries can lead to a considerably more efficient representation of states in the tensor network. This part is intended to be suitable as an implementation-oriented introduction to tensor networks in general and the implementation of non-abelian symmetries in particular. Second, it introduces a series of technical improvements for the MPS methods. These improvements include a faster convergence scheme for MPS-DMRG, a systematic approach to the construction of matrix-product operators and an improved Krylov time evolution method as well as the combination of several well-known techniques into a single tensor network toolkit, SYTEN. The effectiveness of these improvements is demonstrated in numerical examples. Third, the toolkit is applied to the study of two models of current research interest: A one-dimensional spin chain in a staggered external magnetic field is studied and confinement of the elementary spinon excitations, as predicted by analytical arguments, found numerically using real-time evolution and evaluation of the dynamical structure factor. Additionally, the Hubbard model in two dimensions is studied extensively at various system sizes, geometries, interaction strengths U and filling factors n using up to 30'000 SU(2)-Spin-symmetric states equivalent to approx. 100'000 states in other MPS-DMRG implementations. Hints of a possible phase coexistence in the region 0.85 < n < 0.95 are found at intermediate interaction strengths U = 4 and U = 6 as well as a consistently striped ground state in the region n ≈ 0.875.
Emergent phenomena of interacting quantum many-body systems arguably pose some of the greatest challenges to modern theoretical physics. Many of these systems are characterized by drastic changes in their properties as they are driven into a regime where quantum effects and interactions become relevant. Examples are the fractional quantum Hall effect and high-temperature superconductors, to name just two. New theoretical tools are necessary to study such systems, as even simple model Hamiltonians are not fully understood after decades of intense research. Numerical simulations are emerging as an indispensable tool alongside more traditional analytic approaches and have seen immense progress in recent years. In this thesis, we study tensor network states, a new class of numerical methods that have been developed in recent years drawing on insights from quantum information theory. Combining renormalization group methods with knowledge about the entanglement structure of ground states of quantum systems, tensor network states aim to efficiently describe ground states of strongly correlated quantum systems. Center stage is taken by the projected entangled-pair states. This ansatz for two-dimensional systems is discussed in detail and its capabilities are assessed by comparison to established methods and by exploring a frustrated system. We then develop a general formalism to exploit Abelian symmetries in a tensor network algorithm and demonstrate its application to projected entangled-pair states. A large part of this thesis is devoted to the discussion of a class of supersymmetric models for interacting lattice fermions. After introducing the concept of supersymmetry in quantum mechanics and an extensive overview of conformal field theory and its relation to numerical simulations, we study the critical theory of the supersymmetric model on the square ladder. We use two tensor network state algorithms, the density-matrix renormalization group and multi-scale entanglement renormalization along with exact diagonalization to explore this challenging system. In the final two chapters, we discuss two applications of tensor network states, namely simulations of the SU(3) Heisenberg model in two dimensions using two different tensor network state algorithms with the goal of illuminating the nature of its ground state, and the indistinguishability, a measure to detect phase transitions and classify wave functions which is particularly suitable for calculations based on tensor network states.
We present a review of the method we have elaborated to compute the correlation functions of the
XXZ spin-1/2 Heisenberg chain.This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz bamework, and leads to a mul- tiple integral representation of the dynamical correlation functions.We describe in particular some recent advances concerning the two-point functions: in the finite chain, they can be expressed in terms of a single multiple integral.Such a formula provides a direct analytic connection between the previously obtained multiple integral representations and the form factor expansions for the correlation functions.