In this, the third paper in our series describing the excitations of the higher-lattice gauge theory model for topological phases, we will examine the (3+1)-dimensional case in detail. We will explicitly construct the ribbon and membrane operators which create the topological excitations, and use these creation operators to find the pattern of condensation and confinement. We also use these operators to find the braiding relations of the excitations, and to construct charge measurement operators which project to states of definite topological charge. Published by the American Physical Society 2024
In this, the third paper in our series describing the excitations of the higher-lattice gauge theory model for topological phases, we will examine the (3+1)-dimensional case in detail. We will explicitly construct the ribbon and membrane operators which create the topological excitations, and use these creation operators to find the pattern of condensation and confinement. We also use these operators to find the braiding relations of the excitations, and to construct charge measurement operators which project to states of definite topological charge. Published by the American Physical Society 2024
In this series of papers, we study a Hamiltonian model for (3+1)-dimensional topological phases introduced in [Bullivant et al., Phys. Rev. B 95, 155118 (2017)], based on a generalization of lattice gauge theory known as ``higher-lattice gauge theory.'' Higher-lattice gauge theory has so-called ``2-gauge fields'' describing the parallel transport of lines, in addition to ordinary 1-gauge fields which describe the parallel transport of points. In this series we explicitly construct the creation operators for the pointlike and looplike excitations supported by the model. We use these creation operators to examine the properties of the excitations, including their braiding statistics. These creation operators also reveal that some of the excitations are confined, costing energy to separate that grows linearly with the length of the creation operator used. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also discuss the topological charges of the model and use explicit measurement operators to rederive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the model on the 3-torus. From these measurement operators, we can see that the ground-state degeneracy on the 3-torus is related to the number of types of linked looplike excitations. This first paper provides an accessible summary of our findings, with more detailed results and proofs to be presented in the other papers in the series.
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.
Topological phases of matter are characterised by long-range entanglement between their constituent degrees of freedom, which allows them to host excitations with non-trivial exchange statistics. While topological phases in 2+1d are relatively well studied, less is known about their 3+1d counterparts. In this thesis, we examine features of these higher-dimensional phases in a tractable example model, based on a generalisation of lattice gauge theory called higher lattice gauge theory. As well as hosting point-like excitations, the model supports loop-like excitations with non-trivial loop-loop and point-loop braiding statistics. We explicitly construct operators to produce and move these excitations, and use these to find the loop-loop and point-loop braiding relations. These creation operators also reveal that some of the excitations are confined, costing energy to separate, while others are condensed and can be produced locally. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also explain a method for measuring topological charge and use explicit measurement operators to re-derive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the higher lattice gauge theory model on the 3-torus. From these measurement operators, we can see that the ground state degeneracy on the 3-torus is related to the number of types of linked loop-like excitations.
(3+1)-dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for nontrivial loop-loop and point-loop braiding statistics not permitted to point-like excitations alone. Furthermore, these loop-like excitations can be linked together, changing their properties. In particular, this can lead to distinct three-loop braiding, involving two loops undergoing an exchange process while linked to a third loop. In this work, we investigate the loop-like excitations in a (3+1)-dimensional Hamiltonian realization of Dijkgraaf-Witten theory through direct construction of their membrane operators, for a general finite Abelian group and 4-cocycle twist. Using these membrane operators, we find the braiding relations and fusion rules for the loop-like excitations, including those linked to another loop-like excitation. Furthermore, we use these membrane operators to construct projection operators that measure the topological charge and show that the number of distinct topological charges measured by the 2-torus matches the ground-state degeneracy of the model on the 3-torus, explicitly confirming a general expectation for topological phases. This direct construction of the membrane operators sheds significant light on the key properties of the loop-like excitations in (3+1)-dimensional topological phases.
Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be ``twisted,'' in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.
Microscopic Constructions of the BF+AAB Topological Field Theory and Borromean-Rings Braiding in (3+1) Dimensions
Topological order was first studied in fractional quantum Hall states, where anyons reflect electron fractionalization. Recent advances in continuum topological field theories have revealed braiding, fusion, shrinking, and diagrammatic consistency relations in higher dimensions [e.g., Huang, Zhang, Ye, JHEP 11 (2023) 210; JHEP 06 (2025) 238]. As the next step of research, these advances call for explicit microscopic lattice constructions that faithfully reproduce the structures and processes described by continuum field theories. Here we construct, step by step, microscopic counterparts of the (3+1)-dimensional BF field theory with an AAB twist and gauge group (Z2)3. This field theory was introduced in our earlier work on Borromean-Rings braiding [Chan, Ye, Ryu, Phys. Rev. Lett. 121, 061601 (2018)]. We start with the construction of string and thickened membrane operators that create, fuse, shrink, and braid particle and loop excitations, providing microscopic counterparts of Wilson operators in continuum field theories. We derive fusion and shrinking rules and the operators governing particle--loop and Borromean-Rings braiding. We demonstrate selective control of non-Abelian shrinking channels through the internal degrees of freedom of loop operators and establish fusion--shrinking consistency for arbitrary finite groups. An explicit correspondence between all 22 excitations of the D4 quantum double lattice model and the BF+AAB theory preserves fusion and shrinking rules and particle--loop and Borromean-Rings braiding phases. These constructions place the continuum results on a microscopic lattice footing with tensor-product local Hilbert spaces and short-range interactions.