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.
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.
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.
Topological phases of matter are characterized for having a topologically dependent ground state degeneracy, anyonic quasi-particle bulk excitations and gapless edge excitations.Different topologically ordered phases of matter can not be distinguished by te usual Ginzburg-Landau scheme of symmetry breaking.Therefore, a new mathematical framework for the study of such phases is needed.In this dissertation we present the simplest example of a topologically ordered system, namely, the Toric Code (TC) introduced by A. Kitaev in [1].Its ground state is 4-fold degenerate when embedded on the surface of a torus and its elementary excited states are interpreted as quasi-particle anyons.The TC is a particular case of a more general class of lattice models known as Quantum Double Models (QDMs) which can be interpreted as an implementation of (2 + 1) Lattice Gauge Theories in the Hamiltonian formulation with discrete gauge group G.We generalize these models by the inclusion of matter fields at the vertices of the lattice.We give a detailed construction of such models, we show they are exactly solvable and explore the case when the gauge group is set to be the abelian Z 2 cyclic group and the matter degrees of freedom to be elements of a 2-dimensional vector space V 2 .Furthermore, we show that the ground state degeneracy is not topologically dependent and obtain the most elementary excited states. ResumoFases topológicas da matéria são caracterizadas por terem uma degenerescência do estado fundamental que depende da topologia da variedade em que o sistema físico é definido, além disso apresentam estados excitados no interior do sistema que são interpretados como sendo quase-partículas com estatística de tipo anyonica.Estes sistemas apresentam também excitações sem gap de energia em sua borda.Fases topologicamente ordenadas distintas não podem ser distinguidas pelo esquema usual de quebra de simetria de Ginzburg-Landau.Nesta dissertação apresentamos como exemplo o modelo mais simples de um sistema com Ordem Topológica, a saber, o Toric Code (TC), introduzido originalmente por A. Kitaev em [1].O estado fundamental deste modelo apresenta degenerescência igual a 4 quando incorporado à superfície de um toro.As excitações elementares são interpretadas como sendo quase-partículas com estatística do tipo anyonica.O TC é um caso especial de uma classe mais geral de models chamados de Quantum Double Models (QDMs), estes modelos podem ser entendidos como sendo uma implementação de Teorias de gauge na rede em (2 + 1) dimensões na formulação Hamiltoniana, em que os graus de liberdade vivem nas arestas da rede e são elementos do grupo de gauge G. Nós generalizamos estes modelos com a inclusão de campos de matéria nos vértices da rede.Também apresentamos uma construção detalhada de tais modelos e mostramos que eles são exatamente solúveis.Em particular, exploramos o modelo que corresponde à escolher o grupo de gauge como sendo o grupo cíclico Z 2 e os graus de liberdade de matéria como sendo elementos de um espaço vetorial bidimensional V 2 .Além disso, mostramos que a degenerescência do estado fundamental não depende da topologia da variedade e obtemos os estados excitados mais elementares deste modelo.
We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+1 dimensions, based on a generic finite group G and a 4-cocycle $\ensuremath{\omega}$ over G. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3,Z) generators as the modular S and T matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the (3+1)-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group G. This work is presented to be accessible for a wide range of physicists and mathematicians.
We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group G and a 3-cocycle $\ensuremath{\alpha}$ over G. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of G. When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group G, in which the elementary excitations are known to be classified by the quantum double D(G) of G. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net 6j symbols to the corresponding 3-cocycles.
Over the past 30 years experimental observations have demonstrated the existence of a variety of quantum phases of matter not admissible to a classification in terms of the Landau theory of symmetry breaking. Examples include, but are not limited to the fractional quantum
Hall states and frustrated quantum magnets. Theoretical evidence supports the idea that such phases can exist in a large class of zero temperature strongly correlated condensed matter systems.
In this thesis we study a particular case of such systems called topological phases of matter. Such phases are characterised by the presence of non-local correlations which are manifest in properties such as degenerate groundstates that depend on the global topology of the
system and the emergence of topological excitations. Remarkably the classification of such materials is profoundly tied to the mathematical construction of topological quantum field theories (TQFT).
In this thesis we utilise this connection to explore possible candidate Hamiltonian models for topological phases of matter. Our methodology is that of reverse engineering effective local Hamiltonians from a class of discrete TQFT’s called state sums.
In chapter 5 we develop a construction to canonically associate to any state-sum TQFT a corresponding local Hilbert space and Hamiltonian defining a candidate model for a topological phase of matter.
In chapter 9 we develop a candidate model of topological phases using ideas from higher gauge theory and higher category theory. In particular we define a Hamiltonian realisation of the Yetter Homotopy 2-type TQFT which describes a topological gauge theory, where the
gauge symmetry is given by a finite 2-group and relate a class of such models to the construction of Walker-Wang.
Building on the Hamiltonian construction for state-sum TQFT’s, in Part III of this thesis we develop an algebraic approach to understanding the topological excitations of such theories, we call tube-algebras. In chapter 10 we develop a general construction for defining tube-algebras for any unitary state-sum TQFT and describe the general features. In chapter 12 we apply this construction to the
Dijkgraaf-Witten TQFT in 1+1, 2+1 and 3+1D. In chapter 13 we
apply this construction to topological higher lattice gauge theories and compare the results the Dijkgraaf-Witten TQFT.
We propose an exactly solvable Hamiltonian for topological phases in 3+1 dimensions utilizing ideas from higher lattice gauge theory, where the gauge symmetry is given by a finite 2-group. We explicitly show that the model is a Hamiltonian realization of Yetter's homotopy 2-type topological quantum field theory whereby the ground-state projector of the model defined on the manifold M3 is given by the partition function of the underlying topological quantum field theory for ${M}^{3}\ifmmode\times\else\texttimes\fi{}[0,1]$. We show that this result holds in any dimension and illustrate it by computing the ground state degeneracy for a selection of spatial manifolds and 2-groups. As an application we show that a subset of our model is dual to a class of Abelian Walker-Wang models describing 3+1 dimensional topological insulators.