In 2+1 dimensions, gravity is an SU(2) topological gauge theory that can be written as BF theory. In the condensed matter literature, the Hamiltonian realization of BF theory for finite groups is known as the Kitaev model. The corresponding Hamiltonian yields magnetic and electric point-like excitations, both supported by punctures. In this context, the cylinder plays a special role since the gluing of two cylinders results in another cylinder, hence defining an algebra on the Hilbert space of states, referred to as Ocneanu's tube algebra. By choosing specific graph-states on the cylinder, we can confirm explicitly that this algebra is equivalent to the Drinfel'd double of the gauge group. The representation theory of the Drinfel'd double can then be used to define a basis of excited states associated with any punctured Riemann surface. This result can be adapted so as to define a new basis for the gauge invariant Hilbert space of lattice gauge theories, and a fortiori loop quantum gravity, replacing the well-known spin network basis. In doing so, we naturally shift the focus from the underlying lattice to the excitations themselves, this notion of excitations being understood with respect to a given vacuum state, namely the so-called BF vacuum. This basis diagonalizes so-called ribbon operators that provide Dirac observables. Furthermore, it has an inherent hierarchical organization allowing to design states with a multi-scale structure. This turns out to be extremely precious for studying the large scale structure of the theory. It also provides a new notion of subsystems for gauge theories. Being solely based on excitations, this leads to a completely relational way of defining regions. This can be used to define a new notion of entanglement entropy for lattice gauge theories and (2+1)d gravity coupled to point particles. These techniques can also be generalized so as to define excitation bases for (3+1)d gauge models of topological phases. Two approaches are considered in this thesis. The first one consists in generalizing Ocneanu's tube algebra by replacing the cylinder with the manifold obtained by cutting open a three-torus along one direction. This defines an algebraic structure extending the Drinfel'd double that can be used in a similar fashion as in (2+1)d. The other approach relies on Heegaard splittings of three-manifolds which perform a decomposition into handle-bodies along so-called Heegaard surfaces. We use this technique to encode the Hilbert space of flat connections with curvature excitations of a three-manifold into the Hilbert space of flat connections on a 2d Heegaard surface, hence making the results derived in (2+1)d available to the study of the (3+1)d case.
In 2+1 dimensions, gravity is an SU(2) topological gauge theory that can be written as BF theory. In the condensed matter literature, the Hamiltonian realization of BF theory for finite groups is known as the Kitaev model. The corresponding Hamiltonian yields magnetic and electric point-like excitations, both supported by punctures. In this context, the cylinder plays a special role since the gluing of two cylinders results in another cylinder, hence defining an algebra on the Hilbert space of states, referred to as Ocneanu's tube algebra. By choosing specific graph-states on the cylinder, we can confirm explicitly that this algebra is equivalent to the Drinfel'd double of the gauge group. The representation theory of the Drinfel'd double can then be used to define a basis of excited states associated with any punctured Riemann surface. This result can be adapted so as to define a new basis for the gauge invariant Hilbert space of lattice gauge theories, and a fortiori loop quantum gravity, replacing the well-known spin network basis. In doing so, we naturally shift the focus from the underlying lattice to the excitations themselves, this notion of excitations being understood with respect to a given vacuum state, namely the so-called BF vacuum. This basis diagonalizes so-called ribbon operators that provide Dirac observables. Furthermore, it has an inherent hierarchical organization allowing to design states with a multi-scale structure. This turns out to be extremely precious for studying the large scale structure of the theory. It also provides a new notion of subsystems for gauge theories. Being solely based on excitations, this leads to a completely relational way of defining regions. This can be used to define a new notion of entanglement entropy for lattice gauge theories and (2+1)d gravity coupled to point particles. These techniques can also be generalized so as to define excitation bases for (3+1)d gauge models of topological phases. Two approaches are considered in this thesis. The first one consists in generalizing Ocneanu's tube algebra by replacing the cylinder with the manifold obtained by cutting open a three-torus along one direction. This defines an algebraic structure extending the Drinfel'd double that can be used in a similar fashion as in (2+1)d. The other approach relies on Heegaard splittings of three-manifolds which perform a decomposition into handle-bodies along so-called Heegaard surfaces. We use this technique to encode the Hilbert space of flat connections with curvature excitations of a three-manifold into the Hilbert space of flat connections on a 2d Heegaard surface, hence making the results derived in (2+1)d available to the study of the (3+1)d case.
We consider an exactly solvable model in 3+1 dimensions, based on a finite group, which is a natural generalization of Kitaev’s quantum double model. The corresponding lattice Hamiltonian yields excitations located at torus-boundaries. By cutting open the three-torus, we obtain a manifold bounded by two tori which supports states satisfying a higher-dimensional version of Ocneanu’s tube algebra. This defines an algebraic structure extending the Drinfel’d double. Its irreducible representations, labeled by two fluxes and one charge, characterize the torus-excitations. The tensor product of such representations is introduced in order to construct a basis for (3+1)d gauge models which relies upon the fusion of the defect excitations. This basis is defined on manifolds of the form Σ×S1 , with Σ a two-dimensional Riemann surface. As such, our construction is closely related to dimensional reduction from (3+1)d to (2+1)d topological orders.
Gauge theories and topological phases play a fundamental role in different areas of physics. The first ones are at the basis of the Standard Model in the field of particle physics, describing the electroweak and strong interactions through a non-Abelian gauge theory. In condensed matter and statistical physics, gauge theories arise as low-energy effective descriptions of strongly correlated phenomena, such as quantum spin liquids and quantum Hall effect. In this realm there is a strong connection with topological phases and order, as emergent gauge fermions and bosons often describe collective excitations of new exotic states of matter of spin models. The discretization on a lattice is a possible way of dealing with the strongly coupled nature of these theories, due to the fact that this formulation at finite volume provides natural regularizing cut-offs, i.e. the lattice size and spacing. This allows for the investigation of different non-perturbative properties both numerically and analytically. Despite the success of these methods, there are various aspects which remain intractable due to the sign or complex action problems, like the out of equilibrium real time evolution or the analysis of quantum chromodynamics with finite chemical potential. In this respect, quantum simulators of many-body systems to simulate high energy physics arise as promising alternatives to face these problems in the near future. They are quantum systems that can be controlled and used to simulate more complicated systems, whose properties could not be analysed with classical computational, experimental or theoretical tools. In the last decades, there has been a huge development in the fields of quantum optics and atomic physics, allowing for the realization highly precise and controllable platforms by means of trapped ions, superconducting circuits, Rydberg atoms and ultracold atoms in optical lattices. This work of Thesis is part of these fields and has different purposes, all of them connected with the study of gauge theories and topological phases. Firstly, we want to develop a reformulation of lattice gauge theories in terms of gauge invariant fields, in a way to deal solely with physical variables directly in the action. Among several possible advantages, a crucial point is that this can be particularly helpful for the construction of consistent approximation schemes, such as mean-field theories, in order to understand and capture some of the physical features of the theory and make the mean-field approximation consistent with the Elitzur theorem, stating that a local gauge symmetry can not be spontaneously broken. As a second point, we face the problem of simulating higher dimensional gauge theories with ultracold atoms. One of the challenges in more than one dimension is indeed the realization of plaquette interaction terms in the Hamiltonian of lattice gauge theories: these can be engineered through four correlated hoppings in perturbation theory, or by means of constrained hoppings in the dual formulation. In this respect, our main target is to set up an ultracold atomic platform generating the plaquette term in two dimensions using only two correlated hoppings, and protecting gauge invariance through angular momentum conservation. The last point we address in the Thesis is related to the analysis of particles moving in static background gauge potentials, i.e. when the considered gauge field has no dynamical term in the action of the system. Indeed, the dynamics of quantum particles in the presence of static gauge fields gives rise to intriguing physical phenomena. In particular, using ultracold atom setups, the realization of artificial gauge potentials can be used to investigate the physics of topological semimetals, such as Weyl or Dirac type. In the last ten years, a lot of attention has been paid to their characterization, due to the appearence of clear theoretical predictions and very well-controlled experimental techniques. We investigate here the relation between topological phase transitions and van Hove singularities, i.e. the discontinuities in the energy derivative of the density of states, in three-dimensional gapless systems. In such materials, topological phase transitions can be defined by changes of the topological invariants of the Fermi sheets, happening at specific singular points. Moreover, Fermi surface singularities result in the presence of the so-called van Hove points. We then present a general argument to relate topological phase transitions and van Hove singularities, and show observable consequences that are related to the transport properties of the system. We exemplify our argument in Weyl systems by analyzing the three-dimensional Hofstadter model for various commensurate fluxes, which offers the opportunity to consider different kinds of Weyl metals and to understand the features of their density of states.
Doubled topological phases introduced by Kitaev, Levin, and Wen supported on two-dimensional lattices are Hamiltonian versions of three-dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state is shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three-dimensional geometrical interpretation are given.
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.
Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.
Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.\n\n\n\nThe operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.