Determining the properties of quantum many-body systems is a central challenge in modern physics. Being able to determine the macroscopic properties of a system from its microscopic description would hasten progress in many fields of science and technology. However, we currently lack the tools to solve such problems generally, or even to develop a theoretical intuition about how many systems might behave. From a simple Hamiltonian description of the system, one may obtain complex, highly correlated collective behaviour. Computational techniques have played a major part in the effort to determine properties of quantum many-body systems. However, as the total degrees of freedom in the system scales exponentially in the system size, numerical diagonalization of the Hamiltonian quickly becomes computationally intractable and one must develop more efficient approximate techniques to explore the system. Present numerical methods such as quantum Monte Carlo and series expansion have provided insight into many systems of interest, but are also held back by fundamental difficulties. In this thesis, we focus on tensor networks, a relatively new ansatz for representing quantum many-body states. Tensor networks are motivated by two ideas from quantum information: firstly, that quantum entanglement is the source of the immense difficulty of simulating quantum systems classically, and secondly that the ground states of certain Hamiltonians exist in a low-entanglement region of the entire Hilbert space. The strength of tensor networks is that they provide a systematic way of representing this class of low-entanglement quantum states. In particular, this thesis describes the iPEPS algorithm for computing the ground states of infinite, two-dimensional quantum lattice systems based on the Projected Entangled Pair States (PEPS) ansatz. We then benchmark the algorithm by computing the phase diagrams of several systems that have been studied with other techniques. Lastly, we apply our algorithm to problems that are not well solved by current approaches, such as frustrated spin systems.
Determining the properties of quantum many-body systems is a central challenge in modern physics. Being able to determine the macroscopic properties of a system from its microscopic description would hasten progress in many fields of science and technology. However, we currently lack the tools to solve such problems generally, or even to develop a theoretical intuition about how many systems might behave. From a simple Hamiltonian description of the system, one may obtain complex, highly correlated collective behaviour. Computational techniques have played a major part in the effort to determine properties of quantum many-body systems. However, as the total degrees of freedom in the system scales exponentially in the system size, numerical diagonalization of the Hamiltonian quickly becomes computationally intractable and one must develop more efficient approximate techniques to explore the system. Present numerical methods such as quantum Monte Carlo and series expansion have provided insight into many systems of interest, but are also held back by fundamental difficulties. In this thesis, we focus on tensor networks, a relatively new ansatz for representing quantum many-body states. Tensor networks are motivated by two ideas from quantum information: firstly, that quantum entanglement is the source of the immense difficulty of simulating quantum systems classically, and secondly that the ground states of certain Hamiltonians exist in a low-entanglement region of the entire Hilbert space. The strength of tensor networks is that they provide a systematic way of representing this class of low-entanglement quantum states. In particular, this thesis describes the iPEPS algorithm for computing the ground states of infinite, two-dimensional quantum lattice systems based on the Projected Entangled Pair States (PEPS) ansatz. We then benchmark the algorithm by computing the phase diagrams of several systems that have been studied with other techniques. Lastly, we apply our algorithm to problems that are not well solved by current approaches, such as frustrated spin systems.
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.
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.
Strongly correlated quantum systems are one of the most interesting and challenging phenomena in modern condensed matter physics. Unfortunately, the theoretical access even to their low energy properties is limited by the exponential growth of the Hilbert space in a many-body setting. To mitigate this computational barrier, tensor network methods are one of the most competitive approaches to classically simulate complex quantum systems. In this thesis we will discuss recent developments in the field of tensor networks to efficiently and accurately simulate two-dimensional quantum many-body states in the thermodynamic limit. Hereby, we introduce the variational update globally optimizing the wave-function, encoded as tensor network in the form of an infinite projected entangled pair state (iPEPS), in order to obtain ground states of many-body Hamiltonian models. We focus in the discussion on the well-established method to approximately contract this two-dimensional structure, the corner transfer matrix renormalization group (CTMRG), and discuss its native application on square lattice and triangular lattice based models. Furthermore, we report the development of a refined numerical algorithm, the split-CTMRG, to significantly reduce its computational complexity, pushing the limits of feasible simulations within modern hardware capabilities. Finally, the method advancements are applied to a currently widely studied model of frustrated magnetism, the Heisenberg maple-leaf antiferromagnet. We are able to find a large phase of vanishing magnetization in contradiction to previous works and, to the best of our knowledge, the lowest reported variational ground state energy at the isotropic point of the model. As an outlook, we describe a potential adaption of the previously developed methods to variationally simulate two-dimensional many-body thermal states.
The efficient simulation of a quantum many-body system is often incredibly difficult due to the exponential scaling of the Hilbert space with increasing number of interacting particles. In practice, it is only possible to exactly simulate small clusters. In many cases, tensor-network-based methods are capable of overcoming this challenge, providing reliable simulations of much larger or even infinite-sized systems through a low-rank factorization of the state coefficients. Tensor operations are typically highly parallelizable, and therefore capable of utilizing the high performance of modern general-purpose graphics processing units (GPU). A particular tensor-network ansatz called infinite projected-entangled pair states (iPEPS) may be used to simulate infinite two-dimensional lattice systems, and are the primary subject of this thesis. As part of this work, a GPU-based open-source software library called Ace-TN was developed to accelerate iPEPS calculations. Benchmarks are presented showing efficient single- and muli-gpu execution of the core iPEPS simulation techniques. iPEPS simulations are performed for quantum spin systems with bond directional interactions, referred to as quantum compass models (QCM) or Kitaev models depending on pre-established naming conventions. The QCM defined on square and kagome lattices are considered with the addition of an external magnetic field. Each system possesses one-dimensional gauge-like symmetries in zero field, relating subextensively-many ground states that are capable of being represented by iPEPS. These states are shown to be metastable in non-zero field through adiabatic evolution from the zero-field ground state, each separated by a small energy gap. Several field-dependent phases and phase transitions are found in each system. Prominent nematic phases are found in both the square- and kagome-lattice systems. The square-lattice system contains a field value that admits an exact solution of the ground state which is characteristic of the surrounding phase. The inclusion of isotropic Heisenberg coupling to the kagome-lattice system is also explored, which produces a frustrated parameter regime between two dual quantum spin-liquid phases.
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.
Tensor network algorithms (TNAs) represent one of the most recent developments in the field of numerical methods for the simulation of strongly-correlated manybody systems.Arising as a natural consequence of an improved understanding of the entanglement structures intrinsic to the ground-state manifolds of manybody Hilbert spaces, they correspond to algorithms exploiting entropic constraints expected to arise in various classes of many-body ground states.In this work we apply and develop TNAs for the simulation of various stronglycorrelated lattice models.Concretely, we apply tensor network renormalization (TNR) to the study of the classical Blume-Capel model (BCM).We propose to exploit the RG features specific to TNR to obtain an indicator for the vicinity of (multi-)critical points.We show that in the case of the BCM it leads to a location of its tricritical point matching the accuracy of state-of-the-art Monte Carlo approaches.This allows us to characterize the underlying c = 7/10 conformal field theory with an excellent accuracy.We then present a self-contained introduction to the most widely used techniques for the simulation of one-and two-dimensional quantum systems, where we cover matrix product states (MPS) and projected entangled-pair states (PEPS) in detail.We briefly discuss the multi-scale entanglement renormalization Ansatz (MERA).We apply infinite PEPS (iPEPS) to the simulation of the Kitaev-Heisenberg (KH) model, proposed as an effective low-energy theory for the so-called Iridate compounds of the form A 2 IrO 3 (A = Na, Li).We show the ability of iPEPS to accurately encode the complex ground-state physics of Kitaev's honeycomb model.When considering the KH model we confirm the existence of all previously found phases, locate all phase transitions in the phase diagram, finding good agreement with previous studies, and provide estimates for the survival regions of the spinliquid phases in the thermodynamic limit.We briefly discuss the nature of these transitions.We conclude this work with a study of various formulations of iPEPS on cylindrical geometries.We benchmark the proposed formulations by studying the transversefield Ising model and find good performance for a subset of the formulations studv ied.We then carry out a comparison between iPEPS and iMPS methods for the Heisenberg and Hubbard models and find a range of cylinder widths over which both methods exhibit comparable performance.We find evidence for the potential of iPEPS simulations on cylinders and argue that our findings provide support for future studies employing both MPS and PEPS methods in conjunction.vi Wir schlieSSen diese Arbeit mit einer Untersuchung verschiedener Formulierungen von iPEPS für zylindrische Geometrien ab.Wir bewerten die vorgeschlagenen Formulierungen durch die Untersuchung des Ising-Modells im transversalen Feld und finden gute Ergebnisse für eine Teilmenge der untersuchten Formulierungen.Wir führen dann einen Vergleich zwischen iPEPS und iMPS Methoden für die Heisenberg und Hubbard Modelle durch und finden einen Bereich von Zylinderbreiten, in dem beide Methoden vergleichbare Leistungen bringen.Wir finden Beweise für das Potenzial von iPEPS-Simulationen auf Zylinder und argumentieren, dass unsere Erkenntnisse Unterstützung für zukünftige Studien liefern, die sowohl MPSals auch PEPS-Methoden zusammen verwenden.viii Despite the considerable progress made towards the understanding of stronglycorrelated systems, much work remains to be done and, as is perhaps to be expected, the broad nature of the problems encountered means that no single approach is likely to be applicable in every circumstance.From a numerical perspective a wide variety of techniques have been developed over time.Of these, the most accurate is exact diagonalization (ED) which, as its name indicates, attempts to partially diagonalize the Hamiltonian operator.However, owing to the exponential growth of Hilbert space dimension as the system size is increased, this method rapidly becomes inapplicable.Indeed, even when targeting