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arXiv:2512.21148

Microscopic Constructions of the $BF+AAB$ Topological Field Theory and Borromean-Rings Braiding in $(3+1)$ Dimensions

Yizhou Huang, Zhi-Feng Zhang, Qing-Rui Wang, Peng Ye

cond-mat.str-elcond-mat.mes-hallhep-thquant-phSubmitted Dec 24, 2025• Announced Dec 25, 2025• 0 citations• Revised Oct 5, 2026 (v5)
v1 · Dec 24, 2025v2 · Feb 17, 2026v3 · Mar 11, 2026v4 · Jun 2, 2026v5 · Oct 5, 2026
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Abstract

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 $(\mathbb{Z}_2)^3$. 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 $\mathbb{D}_4$ 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.

Comments
108 pages

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Microscopic Constructions of the BF+AABBF+AABBF+AAB Topological Field Theory and Borromean-Rings Braiding in (3+1)(3+1)(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)(3+1)(3+1)-dimensional BFBFBF field theory with an AABAABAAB twist and gauge group (Z2)3(\mathbb{Z}_2)^3(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\mathbb{D}_4D4​ quantum double lattice model and the BF+AABBF+AABBF+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.
Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in two dimensions (2D) are always particlelike anyons, those in three dimensions (3D) incorporate not only particles but also loops, spatially nonlocal objects, making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher-dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world sheets of unknotted loops into world lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and antiloop may generate multiple vacua, but fusing a pair of anyon and antianyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions for future exploration, e.g., symmetry enrichment, quantum gates, and topological order of braided monoidal 2-category of 2-group.
Braiding phenomena, from the charge-flux Aharonov-Bohm effect to anyonic statistics in fractional quantum Hall systems, are paradigmatic manifestations of topology in quantum physics. Ordinary mutual braiding between ppp- and qqq-dimensional excitations occurs in d=p+q+2d=p+q+2d=p+q+2 spatial dimensions. In this work, we introduce a universal construction of mutual statistics in the adjacent dimension d=p+q+1d=p+q+1d=p+q+1, applicable to excitations obeying ZN\mathbb Z_NZN​ fusion for arbitrary NNN and all excitation dimensions ppp and qqq. The corresponding invariant is the Berry phase accumulated in a simple 4N4N4N-step microscopic unitary process built from local excitation operators on lattices. This process measures the linking of one excitation with the NNN-fold fusion junction of the other, encompassing particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. We establish the quantization and bilinearity of the invariant and show that its field-theory response is governed by the Bockstein homomorphism, motivating the name Bockstein braiding statistics. Interpreting the excitation operators as open symmetry operators turns the same invariant into a direct microscopic diagnostic of mixed anomalies between symmetries. We demonstrate this diagnostic in a (1+1)D spin chain, where the nontrivial Bockstein braiding phase proves the mixed anomaly between the spin-flip symmetry ∏X\prod X∏X and the nearest-neighbor controlled-ZZZ symmetry ∏CZ\prod CZ∏CZ. We construct explicit (2+1)D and (3+1)D lattice analogs, yielding new anomalous symmetry pairs, and apply the framework to strongly coupled (3+1)D continuum gauge theories. Nontrivial Bockstein braiding rules out a fully symmetric gapped phase, obstructs simultaneous condensation of the two excitations, and implies fractionalization of higher-form symmetries.
Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in three-dimensional (3D) real space focused on either self-statistics, braiding statistics, shrinking rules, fusion rules, or quantum dimensions, it has yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological BFBFBF field theory with twisted terms (e.g., AAdAAAdAAAdA and AABAABAAB) as well as a KKK-matrix BBBBBB term to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous KKK-matrix Chern-Simons theory of two-dimensional topological orders, we present general formulas and systematically show how the KKK-matrix BBBBBB term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. To reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean rings braidings with emergent fermions, and calculate their universal topological data. Together with previous efforts, our paper paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of 3D topological orders. Several interesting future directions are also discussed.
Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the e2m2e^2m^2e2m2-loop in the (4+1)D Z4\mathbb Z_4Z4​ toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of −1-1−1. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by H5(B2G,U(1))H^5(B^2G,U(1))H5(B2G,U(1)). Beyond (4+1)D, the fermionic-loop toric codes form a family of Z2\mathbb Z_2Z2​ topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using Z4\mathbb Z_4Z4​ qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of Z2\mathbb Z_2Z2​ topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that ppp-dimensional excitations in 2p+22p+22p+2 spatial dimensions can support anyonic ppp-brane statistics for only even ppp.
Braiding phases among topological excitations are key data for physically characterizing topological orders. In this paper, we provide a field-theoretical approach towards a complete list of mutually compatible braiding phases of topological orders in (3+1)D spacetime. More concretely, considering a discrete gauge group as input data, topological excitations in this paper are bosonic \particles carrying gauge charges and loops carrying gauge fluxes. Among these excitations, there are three classes of root braiding processes: particle-loop braidings (i.e., the familiar Aharonov-Bohm phase of winding an electric charge around a thin magnetic solenoid), multi-loop braidings [Phys. Rev. Lett. 113, 080403 (2014)], and particle-loop-loop braidings [i.e., Borromean Rings braiding in Phys. Rev. Lett. 121, 061601 (2018)]. A naive way to exhaust all topological orders is to arbitrarily combine these root braiding processes. Surprisingly, we find that there exist illegitimate combinations in which certain braiding phases cannot coexist, i.e., are mutually incompatible. Thus, the resulting topological orders are illegitimate and must be excluded. It is not obvious to identify these illegitimate combinations. But with the help of the powerful (3+1)D topological quantum field theories (TQFTs), we find that illegitimate combinations violate gauge invariance. In this way, we are able to obtain all sets of mutually compatible braiding phases and all legitimate topological orders. To illustrate, we work out all details when gauge groups are ZN1,ZN1×ZN2,ZN1×ZN2×ZN3\mathbb{Z}_{N_1},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}ZN1​​,ZN1​​×ZN2​​,ZN1​​×ZN2​​×ZN3​​, and ZN1×ZN2×ZN3×ZN4\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}\times\mathbb{Z}_{N_4}ZN1​​×ZN2​​×ZN3​​×ZN4​​. Finally, we concisely discuss compatible braidings and TQFTs in (4+1)D spacetime.
(3+1)(3+1)(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)(3+1)(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)(3+1)(3+1)-dimensional topological phases.