It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.
It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.
Strongly correlated systems exhibit phenomena -- such as high-T_c superconductivity or the fractional quantum Hall effect -- that are not explicable by classical and semi-classical methods. Moreover, due to the exponential scaling of the associated Hilbert space, solving the proposed model Hamiltonians by brute-force numerical methods is bound to fail. Thus, it is important to develop novel numerical and analytical methods that can explain the physics in this regime. Tensor Network states are quantum many-body states that help to overcome some of these difficulties by defining a family of states that depend only on a small number of parameters. Their use is twofold: they are used as variational ansatzes in numerical algorithms as well as providing a framework to represent a large class of exactly solvable models that are believed to represent all possible phases of matter. The present thesis investigates mathematical properties of these states thus deepening the understanding of how and why Tensor Networks are suitable for the description of quantum many-body systems. It is believed that tensor networks can represent ground states of local Hamiltonians, but how good is this representation? This question is of fundamental importance as variational algorithms based on tensor networks can only perform well if any ground state can be approximated efficiently in such a way. While any state can be written as a tensor network state, the number of parameters needed for the description might be too large. This is not the case for one-dimensional systems: only a few parameters are required to have a good approximation of their ground states; that, in turn, allows for numerical algorithms based on tensor networks performing well. The situation in two dimensions is somewhat more complicated, but it is known that ground states of local Hamiltonians can be expressed as tensor networks with sub-exponentially many parameters. In the present thesis, we improve on these existing bounds strengthening the claim that the language of tensor networks is suitable to describe many-body systems. Another central question is how symmetries of the system such as translational invariance, time-reversal symmetry or local unitary symmetry can be reflected in tensor networks. This question is important as systems appearing in nature might intrinsically possess certain symmetries; on one hand, understanding these symmetries simplifies the description of these systems. On the other hand, the presence of symmetries leads to the appearance of novel phases -- symmetry-protected topological (SPT) order, -- and tensor networks provide the right language to classify these phases. In one dimension and for certain classes of two-dimensional tensor networks (states generated by so-called injective tensors) it is well understood how symmetries of the state can be described. A general framework, however, has yet to be developed. In the present thesis, we contribute to the development of the theory in two ways. We first investigate the question for injective tensors, and generalize the existing proof for any geometry including the hyperbolic geometry used in the AdS/CFT correspondence. Second, we introduce a class of tensor network states that include previously known examples of states exhibiting SPT order. We show how symmetries are reflected in these states thus deepening the understanding of SPT order in two dimensions.
The combination of quantum effects and interactions in quantum many-body systems can result in exotic phases with fundamentally entangled ground state wavefunctions -- topological phases. Topological phases come in two types, both of which will be studied in this thesis. In topologically ordered phases, the pattern of entanglement in the ground state wavefunction encodes the statistics of exotic emergent excitations, a universal indicator of a phase that is robust to all types of perturbations. In symmetry protected topological phases, the entanglement instead encodes a universal response of the system to symmetry defects, an indicator that is robust only to perturbations respecting the protecting symmetry.Finding and creating these phases in physical systems is a motivating challenge that tests all aspects - analytical, numerical, and experimental - of our understanding of the quantum many-body problem. Nearly three decades ago, the creation of simple ansatz wavefunctions - such as the Laughlin fractional quantum hall state, the AKLT state, and the resonating valence bond state - spurred analytical understanding of both the role of entanglement in topological physics and physical mechanisms by which it can arise. However, quantitative understanding of the relevant phase diagrams is still challenging. For this purpose, tensor networks provide a toolbox for systematically improving wavefunction ansatz while still capturing the relevant entanglement properties.In this thesis, we use the tools of entanglement and tensor networks to analyze ansatz states for several proposed new phases. In the first part, we study a featureless phase of bosons on the honeycomb lattice and argue that this phase can be topologically protected under any one of several distinct subsets of the crystalline lattice symmetries. We discuss methods of detecting such phases with entanglement and without.In the second part, we consider the problem of constructing fixed-point wavefunctions for intrinsically fermionic topological phases, i.e. topological phases contructed out of fermions with a nontrivial response to fermion parity defects. A zero correlation length wavefunction and a commuting projector Hamiltonian that realizes this wavefunction as its ground state are constructed. Using an appropriate generalization of the minimally entangled states method for extraction of topological order from the ground states on a torus to the intrinsically fermionic case, we fully characterize the corresponding topological order as Ising x (p - i p). We argue that this phase can be captured using fermionic tensor networks, expanding the applicability of tensor network methods.
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.
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 most general description of a quantum many-body system is given by a wave- function that lives in a Hilbert space with dimension exponential in the number of particles. This makes it extremely hard to study strongly correlated phenomena like the fractional quantum Hall effect and high-temperature superconductivity. Whenever interactions are sufficiently local and temperature is low, the system does not explore the full Hilbert space, but its ground state resides in the small corner of Hilbert space described by the area law. Containing little entanglement, the states can then be expressed as tensor networks, a family of wavefunctions with a polynomial number of parameters. On the one hand, tensor networks can be used as a variational manifold in nu- merical computations. On the other hand, they allow building model wavefunctions much like locality allows writing down physically realistic Hamiltonians. Besides allowing for an analytical treatment, these models grant access both to the physical and the entanglement degrees of freedom. This is particularly useful in classifying phases of matter. A large number of phases can be explained in terms of Landau’s symmetry-breaking paradigm. This framework, however, is not complete, as exemplified by the existence of phases with intrinsic topological order in two dimensions. It was a major conceptual advance when tensor networks could explain (non-chiral) topological phases as those where the symmetry resides in the entanglement degrees of freedom. The symmetries corresponding to those topological phases act as discrete, finite groups on the virtual degrees of freedom. The purpose of this Thesis is to generalize this program to include other symmetries. We investigate a class of tensor networks with continuous symmetries and find that they cannot describe gapped physics with a unique ground state. The abelian case is found to describe a non-Lorentz invariant phase transition point into a topologically ordered phase. The physics of the non- abelian case is that of a plaquette state that spontaneously breaks the translation symmetry of the lattice. The non-abelian PEPS arises as the ground state of a local parent Hamiltonian whose ground state manifold is completely characterized by the tensor network. In both cases, we find two types of corrections to the entanglement entropy: first there is a correction that is logarithmic in the size of the boundary and independent of the shape. A further correction depends only on the shape of the partition, imposing further restrictions on regions that are suffciently thin. Finally, we investigate symmetries that mix the virtual with the physical degrees of freedom and are furthermore anisotropic. Their physics is described by subsystem symmetry protected topological order. In particular, we focus on the entanglement entropy in the cluster phase and show that there is a universal constant correction to the entropy throughout the phase. This is important in the program of establishing the entanglement entropy as a detection mechanism for topologically ordered phases. We put forward a numerical algorithm to compute the correction and use it to discover a novel phase of matter in which the cluster phase is embedded.
Understanding many-body quantum systems is one of the most challenging problems in contemporary condensed-matter physics. Tensor network representation of quantum states and operators are taking central stage in this pursuit and beyond. They prove to be a powerful numerical and conceptual tool, and indeed a new language altogether. This thesis investigates various aspects of these representations by focusing on two specific problems: the first half of the thesis is devoted to examining how 'stable' a tensor network representation is for two-dimensional quantum states with topological order, and the second half explores the representability of various unitary loop operators with tensor networks. In the numerical usage of the tensor networks, the tensor is varied as to find the representation of the ground states of the given Hamiltonian. In chapter two and three of this thesis we show that such a numerical program for topological phases can be 'ill-posed'. We show that tensor network can be an unstable representation for a topological phase: even an infinitesimal variation in the representation results in the loss of topological order, completely or partially. We diagnose this problem by identifying the exact causes of this instability, and find that it is only tensor variations in certain directions that result in instability, because they result in the condensation of bosonic quasi-particles of the phase. Such unstable variations are characterized by two properties: (1) they can replace a tensor in the tensor network without making the network collapse, and (2) their presence in the network represents the presence of a non-trivial topological charge. We prove that the general tensor representation of all string-net models suffer with such instabilities. We propose an exact mathematical operator to project out all such unstable variations and show its efficacy for a few models by direct calculations. Such an operator can be useful in numerical programs involving such tensor representations. We also point out that such variations play a crucial role in simulating topological phase transitions and their presence can be vital in an accurate simulation. In chapter four and five of this thesis we focus on the representability of unitary loop operators by tensor networks. Such operators not only provide an important tool in the study of dynamical process in one-dimensional systems, but also in understanding and classification of symmetry protected topological phases in two dimensions. To characterize all such operators, we find a necessary and sufficient condition for any loop tensor network operator of a given length to represent a unitary operator. In particular, it is shown that all unitary operators that map local operators to local operators (locality-preserving) can always be represented by a tensor network. Locality-preserving unitary loop operators are classified by a rational index called the GNVW index defined in Ref. [1] which measures how much information 'flows' along the loop. We define Rank-Ratio index for tensor network operators and show that it is completely equivalent to the GNVW index. Therefore, GNVW index of a unitary operator can be easily extracted from its tensor network representation. We find that, other than representing locality-preserving unitary maps, tensor networks can also represent unitary operators that map local operators to global (non-local) operators. These tensor network operators are found to have a long-ranged order similar to tensors that represent topological tensor network states in two dimensions.