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Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1)D with higher gauge symmetry

Bullivant, A, Calçada, M, Kádár, Z, Faria Martins, J, Martin, P

Published May 1, 2020• 67 citations
OpenAlex

Abstract

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.

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Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

Alex Bullivant, Marcos Calçada, Zoltán Kádár +2 more

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.

arXiv:1702.00868•Reviews in Mathematical Physics•Oct 10, 2019•61 citations

Topological phases from higher gauge symmetry in 3+1 dimensions

Alex Bullivant, Marcos Calçada, Zoltán Kádár +2 more

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 ${M}^{3}$ 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.

arXiv:1606.06639•Physical review. B./Physical review. B•Apr 13, 2017•49 citations

Gauge Models of Topological Phases and Applications to Quantum Gravity

Clement Delcamp

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.

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Excitations in the higher-lattice gauge theory model for topological phases. II. The (2+1) -dimensional case

Joe Huxford, Steven H. Simon

In this work, the second paper of this series, we study the $(2+1)$-dimensional version of a Hamiltonian model for topological phases based on higher-lattice gauge theory. We construct the ribbon operators that produce the pointlike excitations. These ribbon operators are used to find the braiding properties and topological charge carried by the pointlike excitations. The model also hosts looplike excitations, which are produced by membrane operators. By considering a change of basis, we show that, in certain cases, some looplike excitations represent domain walls between patches corresponding to different symmetry-related ground states, and we find this symmetry. We also map the higher-lattice gauge theory Hamiltonian to the symmetry-enriched string-net model for symmetry-enriched topological phases described by Heinrich, Burnell, Fidkowski, and Levin [Phys. Rev. B 94, 235136 (2016)], again in a subset of cases.

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Excitation basis for (3+1)d topological phases

Clement Delcamp

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 $$ \varSigma \times {\mathbb{S}}_1 $$ , 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.

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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.
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.
We propose an exactly solvable Hamiltonian for topological phases in 3+13+13+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{M}^{3}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+13+13+1 dimensional topological insulators.
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.
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.
In this work, the second paper of this series, we study the (2+1)(2+1)(2+1)-dimensional version of a Hamiltonian model for topological phases based on higher-lattice gauge theory. We construct the ribbon operators that produce the pointlike excitations. These ribbon operators are used to find the braiding properties and topological charge carried by the pointlike excitations. The model also hosts looplike excitations, which are produced by membrane operators. By considering a change of basis, we show that, in certain cases, some looplike excitations represent domain walls between patches corresponding to different symmetry-related ground states, and we find this symmetry. We also map the higher-lattice gauge theory Hamiltonian to the symmetry-enriched string-net model for symmetry-enriched topological phases described by Heinrich, Burnell, Fidkowski, and Levin [Phys. Rev. B 94, 235136 (2016)], again in a subset of cases.
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 \varSigma \times {\mathbb{S}}_1 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.