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arXiv:2307.09983 · Physical Review Research

Continuum field theory of three-dimensional topological orders with emergent fermions and braiding statistics

Zhifeng Zhang, Qing-Rui Wang, Peng Ye

Published Nov 3, 2023• 9 citations
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Abstract

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 $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of two-dimensional topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ 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.

DOI
10.1103/physrevresearch.5.043111

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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 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.
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.
Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2+1d2+1d2+1d, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in 3+1d3+1d3+1d and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in 3+1d3+1d3+1d. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.
Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory. In particular, topologically ordered phases are topological phases with topological order. Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations. For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons. In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations. These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation). The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases. One important aspect is to understand the braiding statistics in topologically ordered phases. While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear. In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase. In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics. It is a particle-loop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase. Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops. Another important aspect is to probe the topological order in topologically ordered phases. One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces. For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons. Also, ground state transformation under twisting the torus encodes the braiding data. In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces. We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics. They in turn distinguish different topological order. Even without topological order, there can be non-trivial topological phases in the presence of global symmetry. These topological phases are called the symmetry protected topological phases. For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological insulators. While the bulk of the topological insulators behaves like trivial insulators, the surface is anomalous with odd number of massless dirac fermions. While most of the understanding of topological insulators is developed upon the free fermion picture, such phase survives under interaction. In the thesis, we derive hydrodynamic field theory for the topological insulators by functional bosonization. Such formulation is robust against interaction. In addition, by suitably incorporating electron interactions, we predict new topological phases, called the fractional topological insulators. Besides, the surface theory of topological insulators also justify the famous duality between free massless dirac fermion and QED_3.
Braiding statistics of ppp- and qqq-dimensional topological excitations is conventionally defined in p+q+2p+q+2p+q+2 spatial dimensions. We find a novel statistical process WN(X,Y)=(Y−1X−1)N(YX)NW_N(X,Y)=(Y^{-1}X^{-1})^N(YX)^NWN​(X,Y)=(Y−1X−1)N(YX)N for two order-NNN excitations in p+q+1p+q+1p+q+1 dimensions, detecting the Bockstein response A⌣β(B)A\smile β(B)A⌣β(B). This new statistics and fermionic loop statistics exhaust all loop statistics in three dimensions whose fusion rules form an Abelian group GGG, classified by H5(B2G,U(1))H^5(B^2G,U(1))H5(B2G,U(1)). Surprisingly, conventional three-loop braiding goes beyond this classification, so it must have non-Abelian fusion rules. We suggest viewing three-loop braiding as particle-loop braiding together with exotic fusion rules between loops and point-like defects. We also try to clarify the relationship between statistics and symmetry anomaly.