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arXiv:1305.0214 · Physical Review B

Changing topology by topological defects in three-dimensional topologically ordered phases

Andrej Mesaroš, Yong Baek Kim, Ying Ran

Published Jul 30, 2013• 36 citations
arXivHTMLPDF

Abstract

A hallmark feature of topologically ordered states of matter is the dependence of ground-state degeneracy (GSD) on the topology of the manifold determined by the global shape of the system. Although the topology of a physical system is practically hard to manipulate, recently, it was shown that in certain topologically ordered phases, topological defects can introduce extra topological GSD. Here the topological defects can be viewed as effectively changing the topology of the physical system. Previous studies have been focusing on two spatial dimensions with pointlike topological defects. In three dimensions, linelike topological defects can appear. They are closed loops in the bulk that can be linked and knotted, effectively leading to complex three-dimensional manifolds in certain topologically ordered states. This paper studies the properties of such line defects in a particular context: the lattice dislocations. We give an analytical construction, together with support from exact numerical calculations, for the dependence of the GSD on dislocations of certain doubled versions of the exactly solvable Kitaev's toric code models in both two and three dimensions. We find that the GSD of the 3D model depends only on the total number of dislocation loops, no matter how they are linked or knotted. The results are extended to ${\mathbb{Z}}_{n}$ generalizations of the model. Additionally, we consider the phases in which the crystalline orders are destroyed through proliferation of double dislocations. The resulting phases are shown to host topological orders described by non-Abelian gauge theories.

DOI
10.1103/physrevb.88.035141

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A hallmark feature of topologically ordered states of matter is the dependence of ground-state degeneracy (GSD) on the topology of the manifold determined by the global shape of the system. Although the topology of a physical system is practically hard to manipulate, recently, it was shown that in certain topologically ordered phases, topological defects can introduce extra topological GSD. Here the topological defects can be viewed as effectively changing the topology of the physical system. Previous studies have been focusing on two spatial dimensions with pointlike topological defects. In three dimensions, linelike topological defects can appear. They are closed loops in the bulk that can be linked and knotted, effectively leading to complex three-dimensional manifolds in certain topologically ordered states. This paper studies the properties of such line defects in a particular context: the lattice dislocations. We give an analytical construction, together with support from exact numerical calculations, for the dependence of the GSD on dislocations of certain doubled versions of the exactly solvable Kitaev's toric code models in both two and three dimensions. We find that the GSD of the 3D model depends only on the total number of dislocation loops, no matter how they are linked or knotted. The results are extended to Zn{\mathbb{Z}}_{n}Zn​ generalizations of the model. Additionally, we consider the phases in which the crystalline orders are destroyed through proliferation of double dislocations. The resulting phases are shown to host topological orders described by non-Abelian gauge theories.
We study a class of three-dimensional (3D) exactly solvable models of topological matter first put forward by Walker and Wang [Front. Phys. 7(2) 150 (2012)]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work, we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground-state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a $\ensuremath{\nu}=\frac{1}{2}$ bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of bFbFbF theories. Our special example of the 3D semion model captures much of the behavior of more general ``confined Walker-Wang models.'' We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk, and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum Hall effect. We find that in these models a fermion is always deconfined in the three-dimensional bulk.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
Developing quantum systems which are robust against noise are of prime importance to the realisation of quantum technologies. Without fault tolerance, we will never be able to preserve delicate quantum states for macroscopic time scales, a necessary requirement in the construction of scalable quantum computer. Topologically ordered models offer very beautiful mechanisms for preserving quantum states. In such models, quantum information is encoded globally over a degenerate topological Hilbert space which offers a natural robustness against local environmental noise. Remarkably, topological order is present in certain realistic condensed-matter systems. This provides a platform for accessing topological order in a laboratory. Moreover, considering condensed-matter systems enables us to combine topological features with other physical effects to enhance their behaviour. In this Thesis we study the physics of topologically ordered lattice models with defects. We seek practical applications of lattice defects for the realisation of a fault-tolerant quantum computer. The first result we present shows that anyonic data of point-like lattice defects, called twists, can be found by measuring the entanglement of the ground state of the host system. The data we learn relates to the capacity of a twist defect to perform quantum computational tasks. The second result in this Thesis concerns the dynamics of the qudit toric code model with line-like defects coupled to a thermal bath. We show we can entropically inspire fragile glassy dynamics in the system. Such dynamics qualitatively improve the coherence times of quantum information encoded in the ground state of the lattice. A novel way of achieving fault-tolerant quantum computation is by producing and manipulating twist defects of topologically ordered systems. In many ways, this paradigm of topological quantum computation is analogous to quantum computation using anyons. The first area of study in this Thesis extends the analogy between anyons and twists. Specifically, we show that the anyonic data of twists in Kitaev's toric code model can be extracted using topological entanglement entropy calculations in the same way as the same data can be extracted from anyons. We show this using a rigorously solvable lattice model as an example to produce exact analytic results. In particular, our results show we can extract the quantum dimension of a twist, and that we can study the quantum dimension of their fusion products. We compare the obtained results with the anyonic data of the Ising anyon model, further probing an analogy drawn previously in the literature. The second result presented in this Thesis shows that the application of lattice defects can introduce novel dynamics to a two-dimensional topologically ordered quantum memory where excitations carry different masses. A two-dimensional topological model which supports an anyon model with a splitting structure allowed by its fusion channels should be able to entropically achieve high-energy excitation configurations before quantum information encoded in its ground state decoheres. We introduce a grid of lattice defects to a local two-dimensional Hamiltonian model, which, when coupled to a thermal bath, will dynamically steer the excitations into high energy configurations with high probability within a suitable temperature regime. We demonstrate the proliferating dynamics using numerical simulations in a low-temperature regime where we show polynomial improvements in coherence times by increasing the system size for small system sizes, as well as coherence times which scale weakly super exponentially with the inverse temperature of the bath. The dynamics we demonstrate provide the first example of a system which entropically steers excitations into high-energy configurations. The dynamics we demonstrate may lead to the development of experimentally tractable architectures for low dimensional quantum memories. The study of thermal stability in this Thesis requires the development of methods of correcting topologically ordered lattices which have suffered errors. Namely, we require a decoding algorithm for the qudit generalisation of Kitaev's toric code. Further work presented in this Thesis compares rigorously two different decoding algorithms which use renormalisation-group techniques to process classical syndrome information about noise suffered by the topological code. In particular, we study the improvement in their thresholds as the local dimension of its physical systems increase. Our numerical results enable us to analyse and identify the limitations of the different methods of decoding.
Coupled-wire constructions have proven to be useful tools to characterize Abelian and non-Abelian topological states of matter in two spatial dimensions. In many cases, their success has been complemented by the vast arsenal of other theoretical tools available to study such systems. In three dimensions, however, much less is known about topological phases. Since the theoretical arsenal in this case is smaller, it stands to reason that wire constructions, which are based on one-dimensional physics, could play a useful role in developing a greater microscopic understanding of three-dimensional topological phases. In this paper, we provide a comprehensive strategy, based on the geometric arrangement of commuting projectors in the toric code, to generate and characterize coupled-wire realizations of strongly interacting three-dimensional topological phases. We show how this method can be used to construct pointlike and linelike excitations, and to determine the topological degeneracy. We also point out how, with minor modifications, the machinery already developed in two dimensions can be naturally applied to study the surface states of these systems, a fact that has implications for the study of surface topological order. Finally, we show that the strategy developed for the construction of three-dimensional topological phases generalizes readily to arbitrary dimensions, vastly expanding the existing landscape of coupled-wire theories. Throughout the paper, we discuss Zm{\mathbb{Z}}_{m}^{\phantom{\rule{0.16em}{0ex}}}Zm​ topological order in three and four dimensions as a concrete example of this approach, but the approach itself is not limited to this type of topological order.
Topological order is a new field of research involving exotic physics. Among other things it has been suggested as a means for realising fault-tolerant quantum computation. Topological degeneracy, i.e. the ground state degeneracy of a topologically ordered state, is one of the quantities that have been used to characterize such states. Topological order has also been suggested as a possible quantum information storage. We study two-dimensional lattice models defined on a closed manifold, specifically on a torus, and find that these systems exhibit topological degeneracy proportional to the genus of the manifold on which they are defined. We also find that the addition of lattice dislocations increases the ground state degeneracy, a behaviour that can be interpreted as artificially increasing the genus of the manifold. We derive the fusion and braiding rules of the model, which are then used to calculate the braiding properties of the dislocations themselves. These turn out to resemble non-abelian anyons, a property that is important for the possibility to achieve universal quantum computation. One can also emulate lattice dislocations synthetically, by adding an external field. This makes them more realistic for potential experimental realisations.
Abstract In this work, we explore a microscopic realization of three types of anyonic symmetries in a Z 2 × Z 2 topological order, corresponding to a double toric code. These symmetries act as nontrivial permutations on the anyon labels of the parent state. We consider a setup consisting of two decoupled Wen plaquette models stacked on top of each other and introduce dislocations that modify the Hamiltonian, giving rise to localized twist defects, eventually inducing interactions between the layers. In this context, branch cuts act as sources of anyon permutations when they cross it. We characterize the defects by calculating their quantum dimensions, and we also consider double loop operators around them that allow us to determine the non-Abelian fusion rules between the defects, including when they carry different anyon permutations.