We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.
We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.
We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework is reminiscent of state-sum models and lattice topological quantum field theories, but is formalised and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism represent discrete path integrals in Euclidean space-time. This language is more directly related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We introduce our formalism by simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.
The multiscale entanglement renormalization ansatz (MERA) is argued to provide a natural description for topological states of matter. The case of Kitaev's toric code is analyzed in detail and shown to possess a remarkably simple MERA description leading to distillation of the topological degrees of freedom at the top of the tensor network. Kitaev states on an infinite lattice are also shown to be a fixed point of the renormalization group flow associated with entanglement renormalization. All of these results generalize to arbitrary quantum double models.
We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. For non-Abelian groups, the resulting model supports non-Abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.
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.
We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with Z 2 topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-1/2 lattice to a class of exactly contractible states on spin- S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how local perturbations induce finite-range correlations in this system. This class of tensor networks is readily translated onto any lattice, and we differentiate between the physical consequences of bipartite and non-bipartite lattices on the properties of the corresponding quantum states. We explicitly show this on the hexagonal, square, kagome and triangular lattices.
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.