Non-Abelian three-loop braiding statistics for 3D fermionic topological phases

Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for 3D interacting fermion systems. Most surprisingly, we discover new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with emergent fermionic particles). On the other hand, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also gives rise to an alternative way to classify FSPT phases. We further compare the classification results for FSPT phases with arbitrary Abelian unitary total symmetry Gf and find systematical agreement with previous studies.

Topological phases of quantum matter are a new kind of quantum phases beyond Landau's paradigm.Since the discovery of fractional quantum Hall effect (FQHE), fractionalized statistics of point-like excitations in topological phases has been intensively studied in 2D strongly correlated electron systems.In the past decade, the theoretical prediction and experimental discovery of topological insulator and topological superconductor in 3D systems have further extended our knowledge of topological phases into higher dimensions.As a unique feature, the excitations of 3D topological phases not only contain point-like excitations, but also contain loop-like excitations.Therefore, the fundamental braiding process is not only limited to particle-particle braiding, but is also extended to particle-loop braiding and loop-loop braiding.It is well known that due to topological reasons, pointlike excitations in 3D can only be bosons or fermions.In addition, particle-loop braiding can be understood in terms of Aharonov-Bohm effect and loop-loop braiding is equivalent to particle-loop braiding(one can always shrink one of the loops into a point-like excitation).As a result, for long time people thought there was no interesting fractional statistics in 3D beyond the Aharonov-Bohm effect.Surprisingly, a recent breakthrough pointed out that loop-like excitations can indeed carry interesting fractional statistics via the so-called three-loop braiding process [1][2][3] : braiding a loop α around another loop β, while both are linked to a third loop γ, as shown in Fig. 1.Apparently, such kind of braiding process can not be reduced to the particle-loop braiding due to the linking with a third loop.So far, it has been believed that the three-loop braiding process is the most elementary loop braiding process in 3D.
Another natural question would be: Whether we can use three-loop braiding process to characterize and classify all possible topological phases for interacting fermion systems in 3D?Recent studies on the classification of topological phases for interacting bosonic and fermionic systems in 3D suggest a positive answer to the above question ?? . Bascally, it has been conjectured that all topological phases in 3D can be realized by "gauging" certain underlying symmetry-protected topological (SPT) phases 4,5 .For bosonic systems, the "gauged" SPT states are known as Dijkgraaf-Witten gauge theory, and it has been shown (at least for Abelian gauge groups) that three-loop braiding process of their corresponding flux lines can uniquely characterize and exhaust all Dijkgraaf-Witten gauge theories 6 .For fermionic systems, some particular examples with Abelian three-loop braiding process are also studied recently 7 .However, it is still unclear how to understand general cases.On the other hand, it is well known that in low dimensions (up to 3D), the group cohomology theory [8][9][10] gives rise to a complete classifica-FIG.1.The so-called three-loop braiding process, that is, braiding one loop α around another loop β, while both of them are linked to a third loop γ FIG. 2. Attaching an open Majorana chain onto a pair of linked loops realizes the so-called Ising non-Abelian threeloop braiding process.
In this work, we attempt to systematically understand the three-loop braiding statistics for gauged interacting FSPT systems with general Abelian unitary symmetries.In particular, we discover new types of non-Abelian three-loop braiding statistics that can be only realized in the presence of fermionic particles (accordingly beyond Dijkgraaf-Witten theories).The simplest symmetry group supporting such kind of non-Abelian three-loop braiding process is Z 2 × Z 8 or Z 4 × Z 4 .(More precisely, the corresponding total groups are if we include the total fermion parity symmetry Z f 2 .)A simple physical picture describing the corresponding non-Abelian statistics can be viewed as attaching an open Kitaev's Majorana chain onto a pair of linked flux lines (Z 2 and Z 8 unit flux lines for the former case and two different Z 4 unit flux lines for the latter case).
In 1D, it has been shown that a Majorana chain will carry two protected Majorana zero modes on its open ends 21 .In 2D, it is also well known the vortex(antivortex) of a p+ip topological superconductor can carry a single topological Majorana zero mode.Thus, it is very natural to ask if flux lines in 3D can also carry topological Majorana zero mode or not.Surprisingly, we find that flux lines carrying topological Majorana zero modes must be linked to each other, as shown in Fig. 2. In contrast, if the loops are unlinked, they can never carry Majorana zero modes, and this is because one can always smoothly shrink the flux loops into a point like excitation with a single Majorana zero mode on it.However, as a point like object in 3D can only be boson or fermion, and it is impossible to be an anyon with Ising non-Abelian statistics.
The non-Abelian nature of the new type three-loop braiding process we discovered can be understood as the two-fold degeneracy carried by a pair of linked flux lines, and the braiding statistics between two loops that linked with a third loop should be characterized by a unitary 2 by 2 matrix instead of a simple U (1) phase factor.An alternative way to understand the non-Abelian nature of the three-loop braiding statistics is to use the standard dimension reduction method to deform the 3D lattice model into a 2D lattice model 22 , i.e., by shrinking the z-direction to single lattice spacing such that the flux line along the z-direction can be regarded as a 2D particle which is exactly the Ising non-Ableian anyon 23 with quantum dimension √ 2. Finally, by explicitly working out all the algebraic constraints of three-loop braiding process for fermionic systems(or equivalently, bosonic systems with emergent fermionic particles), we not only uncover new types of Ising non-Abelian three-loop braiding, but also derive a complete classification of 3D FSPT phases with Abelian unitary symmetry.

A. Preliminaries
We begin with the basics of symmetries in fermionic systems and loop braiding statistics in gauged 3D FSPT phase.

Symmetries in interacting fermion systems
Fermionic systems have a fundamental symmetry-the conservation of fermion parity: P f = (−1) N f , where N f is the total number of fermions.The corresponding symmetry group is denoted as Z f 2 = {1, P f }.In the presence of other global on-site symmetries, the total symmetry group G f is the central extension of the bosonic symmetry group G b by the fermion parity Z f 2 , determined by the 2-cocycle ω 2 ∈ H 2 (G b , Z 2 ).In this work, we consider a general Abelian unitary symmetry group of the following form: where N 0 = 2m is an even integer.One can show that any finite Abelian symmetry group in fermionic systems can be written in this form, after a proper isomorphic transformation.
The bosonic symmetry group is expressed as For simplicity, we will mainly consider the case that m and all N i are powers of 2, i.e., where n 0 ≥ 1.When n 0 = 1 (i.e.m = 1), the central extension of G b is trivial; when n 0 ≥ 2, the central extension of G b is nontrivial.This simplification does not exclude any interesting FSPT phases because odd factors of each N µ can be factored out.Moreover, n 2 and n 3 are always trivial if all N i 's are odd integers.Accordingly, neglecting the odd factors, we only lose some BSPT phases, whose classification and characterization are well studied 8 .

Topological excitations and three-Loop braiding in 3D
To study FSPT phases with symmetry group G f , we will gauge the full symmetry.That is, we introduce a gauge field of gauge group G f and couple it to the FSPT system through the minimal coupling procedure (see Refs. 4 and 6 for details of the procedure).The resulting gauged system is guaranteed to be gapped through that procedure, which is actually topologically ordered.It contains two types of topological excitations: (i) Point-like excitations that carry gauge charge.We label them by a vector q = (q 0 , ..., q K ), where q µ is an integer defined modulo N µ .We will use q to denote both the excitation and its gauge charge.This is legitimate because gauge charge uniquely determines charge excitations.Charge excitations are Abelian anyons.Fusing two charge excitations q 1 and q 2 , we obtain a unique charge excitation q = q 1 + q 2 .
(ii) Loop-like excitations that carry gauge flux.We call them vortices, vortex loops or simply loops, and label them by α, β, . . . .The gauge flux carried by loop α is denoted by , where a µ is an integer defined modulo N µ .There exist many loops that carry the same gauge flux, which differ from each other by attaching charges.Unlinked loops are Abelian, however, they may become non-Abelian when they are linked with other loops.Hence, fusion of vortex loops depend on whether they are linked or not.Nevertheless, regardless Abelian or non-Abelian, gauge flux always adds up.General vortex excitations are not limited to simple loops.For example, they may be knots or even more complicated structure.In this work, we only consider simple loops and links of them.So far, properties of loops are enough to characterize gauged FSPT systems.
We need to consider three types of braiding statistics between the loops and charges 1 : First, charge-charge exchange statistics.A charge is either a boson or fermion, depending on the gauge charge it carries.More explicitly, the exchange statistics of charge q is given by That is, when q 0 is odd, it is a fermion.Otherwise, it is a boson.Mutual statistics between charges are always trivial.
Second, charge-loop braiding statistics, which is the Aharonov-Bohm phase given by θ q,α = q • ϕ α (5)   where "•" is the vector dot product.We single out a special class of vortex loops, those carrying the fermion parity gauge flux ϕ = (π, 0, . . ., 0).We denote these fermionparity loops as ξ f .The mutual statistics between charges and fermion-parity loops are simply given by 24 : We notice that the self-exchange statistics of a charge q is equal to Aharonov-Bohm phase θ q,ξ f , which is required by the very definition of fermion parity symmetry.Third, loop-loop braiding statistics.It was shown in Ref. 1 that the fundamental braiding process between loops is the so-called three-loop braiding statistics (Fig. 1): Let α, β, γ be three loop-like excitations.A three-loop braiding is a process that a loop α braids around loop β while both linked to a base loop γ.
On the other hand, if there is no base loop, the twoloop braiding process can always be reduced to chargeloop braiding statistics: Here q α is the absolute charge carried by loop α, which can be obtained by shrinking the loop to a point.Since charge-loop braiding statistics is universal for all FSPT phases with the same symmetry group G f , two-loop braiding is not able to distinguish different FSPT phases.
In the presence of a base loop γ, the notion of absolute charge is not well defined as shrinking loop α to a point will inevitably touch the base loop.Accordingly, three-loop braiding statistics can go beyond Aharonov-Bohm phases, as already demonstrated in many previous works. 1,6,7ile the gauge group G f is Abelian, three-loop braiding process is not limited to be Abelian.As mentioned above, linked loops can be non-Abelian in general, and three-loop process involves linked loops.Let us consider loops α, β, which are linked to the base loop γ.The base loop γ carries gauge flux , where c is an integer vector.Generally speaking, the fusion space between α and β, denoted as V αβ,c , is multi-dimensional (we use this notation because the fusion and braiding process only depend on the gauge flux of the base loop).More explicitly, where loop δ are the possible fusion channels of α and β.Braiding between α and β is a unitary transformation in the fusion space, which in general is not just a phase, but a matrix, leading to non-Abelian three-loop braiding statistics.Similarly to anyons in 2D, one can define fusion multiplicities N δ αβ,c , F -and R-matrices to describe the loop fusion and braiding properties. 6We give more detailed descriptions in Appendix.

Stacking Group Cases Classification
If m is odd and Table I.Classification of 3D FSPT phases with finite unitary Abelian symmetry groups (For simplicity, we only consider symmetry groups Z Nµ with N µ being power of 2, and we assume N i ≤ N j ≤ N k ≤ N l without loss of generality), where m = N 0 /2 and "gcd" means the greatest common divisor.N ij denotes for the greatest common divisor of N i and N j , similarly for N 0ij , N ijk , N 0ijk and N ijkl .

B. Classification of FSPT phases via three-loop braiding statistics
The main purpose of this work is to obtain a classification of 3D FSPT phases via three-loop braiding statistics, and to study non-Abelian three-loop braiding statistics of gauged FSPT phases.We focus on finite Abelian groups of unitary symmetries, which can generally be written as Eq.(1).
We start by defining a set of 3D topological invariants {Θ µ,σ , Θ µν,σ , Θ µνλ,σ } through the three-loop braiding processes (see method section for full details).Our definitions are very similar to those for 2D FSPTs given in Ref. 24, which actually can be related by dimension reduction 6 .Next, we find 14 constraints on {Θ µ,σ , Θ µν,σ , Θ µνλ,σ }, listed in method section.Out of these constraints, 7 follow directly from 2D constraints 24 , while the other 7 are intrinsically 3D.All intrinsically 3D constraints can be traced back to either the 3D Abelian case 7 or 3D non-Abelian bosonic case 6 .Unfortunately, we are not able to prove all the constraints; those we can prove are discussed in Appendix.Finally, by solving the constraints, we obtain a classification of 3D FSPT phases in Table I.The classification group H stack under the stacking operation has the following general form: (9)   where i, j, k, l take values in 1, 2, ..., K, and A, B i , C ij , D ijk , E ijkl are finite Abelian groups.This classification is one of the main results.While it is obtained from a set of partially conjectured constraints, it agrees with all previously known examples.This justifies the validity of the classification.Below we discuss more details for the stacking group structure of the classification results.
According to the stacking group Eq.( 9) for classifying 3D FSPT phases with Abelian total symmetry G f , we can divide the corresponding topological invariants into five categories, such that the topological invariants in each category are independent of those in other categories, i.e. the constraints only relate topological invariants inside each category.The five categories are: where A is the classification group protected by the  3) "Inverse": given an FSPT phase, there exists an inverse phase, such that stacking the two produces the trivial phase.
We believe the the topological invariants are complete for characterizing FSPT phases with Abelian symmetry group G f , and the constraints are complete so that all solutions are physical.Both completenesses are justified by a comparison with the general group super-cohomology method in Appendix.

C. Statistics-type indicators
Our exploration of the classification scheme also uncovers several new kinds of non-Abelian loop braiding statistics, in particular the new kind that involves Majorana zero modes (Fig. 2), which we have briefly mentioned in the introduction.In fact, the correspondence between the layer construction in Refs.18 and 19 and the threeloop braiding statistics data can be extracted.More explicitly, we pick out several special topological invariants, named statistics-type indicators, to indicate non-Abelian loop braiding statistics with different origins: (1) Θ 00i,j = π (m is odd) is the indicator of the non-Abelian statistics in the Majorana-chain layer, which is generated by the loops carrying unpaired Majorana modes, and a loop carrying one Majorana mode is characterized by its quantum dimension √ 2. (2) Θ f i,j = N0i gcd(2,Ni) Θ 0i,j ̸ = 0 (i ̸ = j) is the indicator of the complex fermion layer, where "f " stands for the fermion-parity loop ξ f with gauge flux ϕ ξ f = (π, 0, ...). ( is the indicator of the non-Abelian statistics in the complex fermion layer, which is generated by degeneracies in the complex fermion layer and the relevant loops have integer quantum dimension. (4) Θ ijk,l ̸ = 0 or {Θ f ij,k = 0, Θ 0ij,k ̸ = 0} is the indicator of the non-Abelian statistics in the BSPT layer, which is generated by degeneracies in the BSPT layer and the relevant loops have integer quantum dimension.
We then prove that the first statistics-type indicator Θ 00i,j = π (m is odd) uniquely indicates the Majoranachain layer.To proceed, we need to obtain an explicit expression of the topological invariant Θ µνλ,σ as the following (The definitions we used below are introduced in Appendix: We assume three loops ξ µ , ξ ν , ξ λ all linked to a base loop ξ σ .Mathematically, let the total fusion outcome η of the three loops ξ µ , ξ ν , ξ λ be fixed, and the standard basis is to let ξ ν firstly fuse with ξ µ , then their fusion channel again fuse with ξ λ .We choose the basis of the first local fusion space V ξν ξµ,c to be diagonalized under the braiding of ξ µ around ξ ν , and the braiding of ξ µ around ξ λ is then generally non-diagonalized under this basis, which is expressed as: where B ξν ξµξ λ ,eσ only braids ξ µ around ξ λ , while it depends on ξ ν , as shown in Fig. 3.And B ξµξν ,eσ is redefined in the same basis as B η ξν ξµξ λ ,eσ : which has the same expression as B ξµξν ,eσ : ξν ξµ,c , as though the fusion space is extended, the basis in the extended fusion space should keep diagonalized under the braiding of ξ µ around ξ ν , as shown in Fig. 4.
Then Θ µνλ,σ can be expressed through: where I is the identity matrix in the vector space ).Now we are ready to go back to the proof.For simplicity we can consider m = 1 only, which is due to Z f 2m is isomorphic to Z f 2 ×Z m , and Z m can be absorbed into i Z Ni part of G f .From constraints Eq. (40) and Eq.(51) in method section, Θ iii,j = mΘ 0ii,j = m 2 Θ 00i,j .Therefore when m = 1, we have the relation Θ 00i,j = Θ iii,j .
Firstly we show that the non-Abelian statistics in Majorana-chain layer (i.e. the Ising type statistics) must have Θ 00i,j = π: Do a dimension reduction for the gauged Ising type FSPT system from 3D to 2D by choosing ξ j as the base loop, and condense all the bosonic quasiparticles (as the Ising type statistics is irrelavant to the bosonic matter).The remaining 2D quasiparticles are exactly the Ising anyons: a vortex carrying one majorana mode σ, a fermion ψ and vacuum 1, which satisfies: where we have Θ σσσ = π.And the 3D topological invariants Θ iii,j is exactly equal to the 2D one Θ σσσ after dimension reduction and the condensation of all bosons, i.e.Θ 00i,j = Θ iii,j = Θ σσσ = π.Secondly we show that Θ 00i,j = π corresponds uniquely to the Ising type statistics: From constraint Eq. (40) in method section, when m = 1, Θ 00i,j can only take values 0 or π; when m is even , Θ 00i,j vanishes.We assume the types of non-Abelian statistics in our gauged FSPT system contain only: (1)   Ising type in Majorana-chain layer (2) fermionic type in complex fermion layer (3) bosonic type in BSPT layer.Solving the constraints as listed in Appendix, and examing the generating phases by mapping to 2D model constructions after dimension reduction 24 , we find that to construct the generating phase Θ 00i,j = π, there always exist loops with quantum dimension √ 2, which is the unique property of Ising anyons in the Majorana-chain layer.
The second statistics-type indicator Θ f i,j =  By checking the linear dependence among the topological invariants, we can also determine relations between the three layers, i.e., simply stacked or absorbed.We summarize the group structure of our classification result by layers, i.e., classification corresponds to BSPT phase, complex fermion layer and Majorana chain layer, and whether they are non-trivial group extension (we call absorbed) or simple direct product (we call stacking), in Table II.
Furthermore, invoking the known model construction for 2D FSPT phases 24 and by the fact that quantum dimensions are invariant under dimension reduction, we can find the quantum dimensions of loop-like excitations linked to certain base loops.From the quantum dimensions, we can further show that the non-Abelian three-loop braiding statistics resulting from the Majorana chain layer is due to the unpaired Majorana modes attached to linked loops.Below we will discuss two simplest examples for such kinds of non-Abelian three-loop braiding statistics.
Table II.Layer group structure of the classification group of 3D FSPT phases with finite unitary Abelian symmetry groups (We assume N i ≤ N j ≤ N k ≤ N l without loss of generality).The classification groups of BSPT layer, complex layer and Majorana chain layer are denoted as B, C and K respectively.As the group structure depends on further cases beyond Table I, we list the further cases in column 4. We denote the simple direct product as ×, and non-trivial group extension as ⋉.And the classification groups with non-Abelian braiding statistics are denoted in red color.

D.
Simplest examples for non-Abelian three-loop braiding statistics Firstly, we recall the stacking group classification of FSPT phases: where from Table I we know that: A protected by Z f 2 is trivial, B 1 and B 2 protected by Z f 2 × Z 2 and Z f 2 × Z 8 respectively are trivial, while C 12 protected by Z f 2 ×Z 2 ×Z 8 is nontrivial.Therefore the classification of FSPT phases for the symmetry group is H stack = C 12 .Then we explicitly show the calculation of C 12 : Invoking the known 2D results and combining with the 3D constraints N σ Θ µ,σ = 0, N σ Θ µν,σ = 0, N σ Θ µνλ,σ = 0, the generating phases for the subsets (C1), (C2), (C3), (C4) and (C5) are: where a, b, c, d, e, f, g, h, i are all integers.By the con- . By the constraint Θ ij,0 + Θ oj,i + 4Θ 0i,j = 0, we have a = 0 (mod 2).By the constraint Θ ij,i = −4Θ i,j , we have c = −g (mod 8).By the constraint Θ ij,j = −Θ j,i , we have e = −h (mod 2).Combining all the constraints: ), e = −h (mod 2), i.e. the generating phases are: while all other topological invariants vanish: (Θ 0,j , Θ j,0 , Θ 0j,0 , Θ 00j,0 , Θ j,j , Θ 0j,j , Θ 00j,j ) = (0, 0, 0, 0, 0, 0, 0) ( (Θ ij,0 , Θ 0j,i ) = (0, 0) (25)   Hence in this case the classification is Z 8 × Z 2 , which is a Z 2 complex fermion layer absorbed into a Z 2 × Z 2 BSPT layer, together forming a Z 4 × Z 2 classification, and then a Z 2 Majorana-chain layer again absorbed into the Z 4 ×Z 2 above, as the complex fermion layer indicator is Θ f i,j = Θ 0i,j = − π 2 .Conveniently we can view the "Z 8 " part of the classification being generated by: where Θ i,j = {0, π} correspond to Abelian BSPT phases, Θ i,j = { π 2 , 3 2 π} are Abelian FSPT phases (contain both BSPT layer and complex fermion layer), and Θ i,j = { π 4 , 3 4 π, 5 4 π, 7 4 π} are non-Abelian FSPT phases (contain all BSPT layer, complex fermion layer and Majorana chain layer).Recall that Θ 00i,j = π (m is odd) is the indicator of the Majorana chain layer.And the four non-Abelian FSPT phases all have (Θ 00i,j , Θ 00j,i ) = (π, π), which means that loops ξ i and ξ j each carry one unpaired Majorana mode simultaneously and both have quantum dimension √ 2, which is the origin of the non-Abelian statistics in Majorana chain layer.On the other hand, the "Z 2 " part of the classification is generated by: where Θ j,i = π is a non-trivial BSPT phase, and Θ j,i = 0 is a trivial BSPT phase.We can also understand the 3D braiding statistics by doing a dimension reduction from 3D to 2D and applying the known model construction for 2D generating phases 24 .Firstly we choose ξ j always to be the base loop, and the 2D system after dimension reduction has symmetry Z f 2 × Z 2 , which has only one generating phase (Θ i , Θ 0i , Θ 00i ) = ( π 4 , − π 2 , π), i.e. the subset (C4) in category C.And it can be realized by a two-layer model construction: the first layer a is a charge-2 superconductor with chiral central charge − 1 2 (Ising type), while the second layer b is a charge-2 superconductor with chiral central charge 1 2 (Ising type).The 2D vortex ξ 0 is composited by a unit-flux vortex in layer a and a unit-flux vortex in layer b, which therefore has quantum dimension 2. The 2D vortex ξ i is composited only by a unit-flux vortex in layer b, which therefore has quantum dimension √ 2. As the quantum dimensions of loops are invariant under dimension reudction, we conclude that for non-Abelian FSPT phases, with ξ j all being base loops, loop ξ 0 has quantum dimension 2 and loop ξ i has quantum dimension √ 2. Secondly we choose ξ i always to be the base loop, and the 2D system after dimension reduction has symmetry Z f 2 × Z 8 , which has two generating phases (Θ j , Θ 0j , Θ 00j ) = ( π 8 , π, 0) and (Θ j , Θ 0j , Θ 00j ) = (0, 0, π), where the first one is trivialized to a Z 2 BSPT in 3D, and both constitute the subset (C5) in category C.Only the second generating phase corresponds to non-Abelian statistics and can be realized by a three-layer model construction: the first layer a is a charge-2 superconductor with chiral central charge − 1 2 (Ising type), the second layer b is a charge-8 superconductor with chiral central charge 0 (Abelian layer), and the third layer c is a charge-2 superconductor with chiral central charge 1 2 (Ising type).The 2D vortex ξ 0 is composited by a unit flux in layer a, four times of unit flux in layer b, and a unit flux in layer c, which therefore has quantum dimension 2. The 2D vortex ξ j is composited only by a unit flux in layer b and a unit flux in layer c, which therefore has quantum dimension √ 2. Thirdly we do not specify the base loop, and let the 2D system after dimension reduction have the full symmetry In conclusion, we find that no matter how we do the dimension reduction, the quantum dimensions of the loops coincide, i.e. in our three-loop braiding system with full symmetry Z f 2 × Z 2 × Z 8 , for those non-Abelian FSPT phases, the loop ξ 0 has quantum dimension 2, and loops ξ i and ξ j both have quantum dimension √ 2, which means that loops ξ i and ξ j each carry an unpaired Majorana mode.
Combining all the constraints: ), e = −l (mod 4), i.e. the generating phases are: while all other topological invariants vanish.Hence in this case the classification is simply stacking with a Z 2 "Majorana chain layer absorbed in complex fermion layer", as the complex fermion layer indicator is Θ f i,j = Θ 0i,j = π.
The "Z 2 " part of the classification can be viewed to be generated by: while all other π valued topological invariants are related by the anti-symmetric constraint of Θ µνλ,σ .We do a dimension reduction by always choosing ξ j as the base loop, and the 2D system has symmetry Z f 2 × Z 4 .We find that (Θ 0i , Θ 00i ) = (π, π) is exactly the second generating phase for this 2D FSPT system, which can be realized by a three-layer model construction 24 : the first layer a is a charge-2 superconductor with chiral central charge 3 2 (Ising type), the second layer b is a charge-4 superconductor with chiral central charge −2 (Abelian layer), and the third layer c is a charge-2 superconductor with chiral central charge 1 2 (Ising type).The 2D vortex ξ 0 is composited by a unit flux in layer a, two times of unit flux in layer b, and a unit flux in layer c, which therefore has quantum dimension 2. The vortex ξ i is composited by a unit flux in layer b and a unit flux in layer c, which therefore has quantum dimension √ 2. As the quantum dimensions of the loops are invariant under dimension reduction, and the symmetry groups of ξ i and ξ j are both Z 4 so that it is free to choose which is Z Ni and which is Z Nj , we conclude that in our gauged 3D FSPT systems, loop ξ 0 has quantum dimension 2 and both loop ξ i and ξ j have quantum dimension √ 2. Then we can again check the quantum dimension of loops by doing the dimension reduction without specifying the base loop, and the 2D system has the full symmetry )) can also be realized by a four-layer construction similarly as in the first example.Then the quantum dimension of ξ 0 will still be found as 2, and the quantum dimensions of ξ i and ξ j as both √ 2. Therefore in our construction the nontrivial non-Abelian FSPT phase in the Z 2 classification is due to the unpaired Majorana modes attached on ξ i and ξ j .

III. CONCLUSIONS AND DISCUSSIONS
In summary, we obtain the classification of 3D FSPT phases with arbitrary finite unitary Abelian total symmetry G f , by gauging the symmetry and studying the topological invariants {Θ µ,σ , Θ µν,σ , Θ µνλ,σ } defined through the braiding statistics of loop-like excitations in certain three-loop braiding processes and solving the corresponding constraints for these topological invariants.We further compare this result with the classification obtained by the general group supercohomology theory in Ref 19 and find a systematical agreement.In particular, we can realize any set of allowed values of topological invariants corresponding to a distinguished FSPT phase.Moreover, from several special topological invariants, we can further identify different origins of Non-Abelian three-loop braiding statistics from the corresponding FSPT constructions, i.e., the Majorana chain layer, and complex fermion layer and BSPT layer.Specifically, we argue that the non-Abelian statistics in the Majorana chain layer is due to the unpaired Majorana modes attached on loops.
For future study, it remains unknown how to apply the braiding statistics method to SPT phases with antiunitary symmetry such as the time reversal symmetry, as we do not know how to gauge an antiunitary symmetry.It is expected to generalize the Abelian total symme-try groups G f to general non-Abelian symmetry groups and have a complete understanding of topological invariants for FSPT phases in 3D.Of course, how to use Non-Abelian three-loop braiding statistics to realize topological quantum computation would be anther fascinating future direction.Potential application in fundamental physics was also discussed in Ref. ? , it was conjectured that elementary particles could be further divided into topological Majorana modes attached on linked loops and such a scenario naturally explains the origin of three generations of elementary particles.

A. Definitions of topological invariants
Generally speaking, the full set of braiding statistics among particles and loops is very complicated, in particular when the braiding statistics are non-Abelian.Here, we define a subset of the braiding statistics data, which we call topological invariants.They are Abelian phase factors associated with certain composite threeloop braiding processes, and thereby are easier to deal with.Yet, this subset still contains enough information to distinguish all different FSPT phases, as we will show later.
(i) We define where The quantity θ ξµ,eσ is the topological spin of the loop ξ µ , when it is linked to another loop ξ σ .It is defined as 23 : where R δ ξµξµ,eσ is the R-matrix between two ξ µ loops in the δ fusion channel, and all loops are linked to ξ σ (see Appendix for details).
(ii) We define Θ µν,σ as the phase associated with braiding ξ µ around ξ ν for N µν times, when both are linked to the base loop ξ σ .Here, N µν is the least common multiple of N µ and N ν .In terms of formulas, we have the following expression where B ξµξν ,eσ denotes the unitary operator associated with braiding ξ µ around ξ µ only once, while both are linked to ξ σ , and I is the identity operator.The operator B ξµξν ,eσ can be expressed in term of R matrices, and F matrices if needed, once we choose a basis for the fusion spaces.
For the topological invariants {Θ µ,σ , Θ µν,σ , Θ µνλ,σ } to be well-defined, we need to show that (1) The corresponding braiding processes indeed lead to Abelian phases and (2) the Abelian phases only depend on the gauge flux of the loops, i.e. independent of charge attachments.The proofs are the same as those for the 2D topological invariants {Θ µ , Θ µν , Θ µνλ }, so we do not repeat them here and instead refer the readers to Ref. 24.(The only addition for 3D is that one needs to carry the base loop index σ in every step of the proofs).The reason that the proofs are identical is that the 3D invariants {Θ µ,σ , Θ µν,σ , Θ µνλ,σ } can be related to the 2D invariants {Θ µ , Θ µν , Θ µνλ } by dimension reduction. 6

B. Constraints of topological invariants
The topological invariants {Θ µ,σ , Θ µν,σ , Θ µνλ,σ } should satisfy certain constraints.We claim that they satisfy the following 14 constraints, Eqs. ( 40)-(46) and Eqs. ( 51)-(57).While we are not able to prove all the constraints, we believe they are rather complete.At least, the solutions to these constraints are all realized in the layer construction of FSPT phases (see Appendix).We divide 14 constraints into two groups.
We briefly explain the meaning of the above constraints.For constraint Eq. ( 40), firstly we notice a fact that N copies of the topological invariant Θ µνλ,σ are equivalent to do the braiding process for N copies the type-µ loop, or type-ν, type-λ loop, expressed as: where [N ξ µ ] means N copies the type-µ loop, which can be obtained directly by the definition of Θ µνλ,σ .Then by this fact, the expression mΘ 0µν,σ can be rewritten as 24 : where f is the fermion-parity loop.And constraint Eq. ( 40) illustrates an equivalence Θ f µν,σ = Θ µµν,σ , explicitly proved in the appendix of Ref. 24.Moreover, as the positions of type-µ and type-ν loops are symmetric in Θ f µν,σ , the equality can be extended to Θ ννµ,σ .The constraint Eq. ( 41) simply points out that the type-µ and type-ν loops are symmetric in a three-loop braiding process.The constraints Eq. (42) and Eq. ( 43) are obtained by rearranging the order of certain braiding processes, where the rearrangements give rise to the non-Abelian phase factors Θ µµν,σ and Θ 00i,σ .For constraints Eq. (44) and Eq. ( 45), there are two corollaries relating the type-µ loop and its anti-loop 24 : where µ denotes for the anti-loop ξ µ with gauge flux ϕ ξ µ = −ϕ ξµ .Combining the two corollaries and inducing the definition of Θ µ,σ exactly give constraints Eq. ( 44) and Eq. ( 45).And the constraint Eq. ( 46) obtained by demanding the chiral central charge vanishes for FSPT phases.
Firstly, we argue that the constraints Eqs.(56)(57) proved in Abelian case still hold in non-Abelian case.The constraint Eq. ( 56) is called the cyclic relation.Imagining that we create N µνσ identical three-loop systems with identical fusion channel and identical total charge.By anyon charge conservation, after braiding and fusion, the total charge should still be N µνσ Q link , where Q link is the total charge for a single three-loop system.Then the next step of the proof is similar to the Abelian case 7 , where the difference is that the "vertical" fusions may have multiple fusion channels (differ only by charges).But we do not need to care about the charges attached on the resultant loop after fusion, as finally the total charge should still be N µνσ Q link , by which we fall into the same result as the proof in Ref 7 .And constraint Eq. ( 57) is actually the cyclic relation Eq. (56) divided by half on both sides (mod 2π), which then involves fermionic statistics and hence an intrinsic fermionic constraint.It can be argued that it holds in non-Abelian case in a similar manner.
Then we can rigorously prove the constraints Eq. (C12) to Eq. (C14).The prerequisite to prove them is to assume a 3D "vertical" fusion rule, which naturally gives the linear properties of the topological invariants, explicitly shown in Appendix.
However, the constraints Eq. (51) and Eq. ( 55) are left unproven.For constraint Eq. (51), it is a generalization of the 2D constraint Θ µνλ = sgn( p)Θ p(µ) p(ν) p(λ) , where the 2D version can be easily proved by a Borromean ring configuration 6 .While here we generalize the totally antisymmetric property for the indices of Θ µνλ,σ to the base loop.And the constraint Eq. ( 55) is simply a conjecture, which means that the topological invariant Θ µ,σ vanishes if the two linked loops fall into the same type.
There are several properties for the generally nontrivial fusion rule α × β = δ N δ αβ δ in our gauged FSPT system, where α, β, δ are all loop-like excitations, and the proofs are the same as the bosonic case given in Ref 6 .The properties are: (1) For any fusion channel δ: which means that different fusion channels only differ by their attached charges.
(2) When a loop α is fused with a charge q, there is exactly one fusion outcome: (3) The fusion multiplicity N q αα = 0, 1, where q is any charge in the fusion channels of α and α.
(4) If ϕ α ′ = ϕ α and ϕ β ′ = ϕ β , then there exist charges q 1 and q 2 such that α ′ = α × q 1 , β ′ = β × q 2 and δ ′ = δ × q 1 × q 2 .Appendix B: Some Basic Definitions Define: The fusion space V δ αβ,c that fuses two loops α, β into a single fusion channel δ with base loop c, is a Hilbert space spanned by the set of orthogonal basis 23,25 : which can be simplified as {|αβ, c; δ⟩} as N δ αβ,c is always 1 in our theory.And the full Hilbert space for the fusion of α, β with base loop c is: Accordingly the spliting space for a single fusion channel δ is spanned by the dual basis: Define: Consider a local system involving only two loops α, β both linked to a base loop γ, and their fusion outcome δ is known.The Abelian R-symbol R δ αβ,c that exchanges two loops α, β, during which their fusion channel δ is fixed, is defined as a map 23,25 : which is a basis-dependent pure phase, as |αβ, c; δ⟩ may differ |βα, c; δ⟩ by a gauge transformation.Specially, the R-symbol R δ αα,c : V δ αα,c → V δ αα,c exchanging two identical loops is basis-independent.
Define: The non-Abelian R-symbol R αβ,c is defined as a matrix: which can be diagonalized by choosing a proper basis if there is no other fusion process involved: where δ 1 , δ 2 , ... are all the possible fusion channels of α and β.
Example: For 2D Ising anyons 23 , which contain anyon types {1, σ, ψ}, where X is the Frobenius-Schur indicator, and the Chern number ν is odd (mod 16) for non-Abelian Ising anyons.Define: For the same system above, similarly the Abelian B-symbol B δ αβ,c that braids loop α around β linked to a base loop γ is defined as: which is basis-independent as it maps between the same fusion space.Define: The non-Abelian B-symbol B αβ,c is defined as: which can be diagonalized by choosing a proper basis if there is no other fusion process involved: Example: For 2D Ising anyons, Define: Consider a local system involving three loops α, β, ϵ all linked to a base loop γ, whose total fusion outcome η is known.The F -symbol F η ϵαβ,c that maps between two different fusion ways, is defined as a generally non-diagonalized matrix 23,25 : Example: For 2D Ising anyons, Define: Consider n loops α 1 , α 2 , ..., α n all linked to a base loop γ, where the total fusion outcome η of the n loops is known.Then we define a standard basis in the total fusion space by specifying a particular fusion order 25 .For example, firstly fusing α 1 and α 2 , then fusing the result with α 3 , then fusing the result with α 4 , and so on.The total fusion space can therefore be decomposited as: (B16) which is equivalently expressed by the diagram in Fig. 5.
Define: Consider a local system involving three loops α, β, ϵ all linked to a base loop γ, where the total fusion outcome of the three loops η is known.The R-matrix R η ϵαβ,c that exchanges two loops α, β, while it is diagonalized in the fusion space of ϵ and α, is defined as a generally non-diagonalized matrix 26 : Define: For the same system above, similarly the Bmatrix B η ϵαβ,c that braids loop α around β, while it is diagonalized in the fusion space of ϵ and α, is defined as a generally non-diagonalized matrix: Example: For 2D Ising anyons, Define: The fusion matrix for a loop α linked to a base loop γ is defined as 27,28 : where M is a finite set called superselection sectors, which is the set of all distinguishable particle types in a theory.Define: The quantum dimension of a loop α linked to a base loop γ is defined as the largest eigenvalue of the In order to prove some of the newly involved 3D constraints, we need to consider a new kind of "vertical" fusion in analogy to the original "horizontal fusion", as shown in Fig. 6 (a) and (b).Consider two Hopf-link systems, where the two loops in dfferent systems are in the same type.Then the 3D "vertical" fusion rule has the form: where the "vertical" fusion is denoted as "•".The fusion outcomes have the same flux but different attached charges Q, and the + in (σ 1 + σ 2 ) means only putting two loops together, which applies when fusing the loops or not does not matter as the charges attached on a base loop do not affect the three-loop braiding process.And this "vertical" fusion rule can be understood in a way that the two loops that are about to fuse annihilate at a point (as particle and antiparticle) to vacuum or some charge Q.And if the fusion outcome is a charge Q, it will be attached to the loop after fusion.First we would like to mention that the expression of the topological invariant Θ µν,σ can be further written as: where the fusion channel δ is arbitrary, as the result is the same for all fusion channels 24 , and I is the identity matrix in the fusion space ⊕ δ V δ ξµξν ,c .Then we consider two three-loop systems, where the three loops in different systems are all in the same type as shown in Fig. 7. Specifically, before the "vertical" fusions, we choose to fix the fusion channel for each three-loop system.Thereby the braiding operator for the whole system before "vertical" fusions are: While after the "vertical" fusions, the braiding operator is: where the vertical fusions ξ 1 µ,σ1 • ξ 2 µ,σ2 = ξ ′ µ,(σ1+σ2) + ξ ′′ µ,(σ1+σ2) +... and ξ 1 ν,σ1 •ξ 2 ν,σ2 = ξ ′ ν,(σ1+σ2) +ξ ′′ ν,(σ1+σ2) +... both generally have multiple fusion outcomes.According to the 4th property in section I, the fusion outcomes of the two loops after "vertical" fusions ξ µ,(σ1+σ2) and ξ ν,(σ1+σ2) are also multiple.And we can choose a particular basis in the fusion space such that the braiding operator is diagonalized.
Then we do the braiding processes for both cases (before and after "vertical" fusions) for N µν times, we obtain an equation: where σ ) is any of the diagonalized entry in the matrix (C4).The eqn. (C5) is equivalent to the claim that: The N µν times of braiding as a whole commutes with the "vertical" fusions.
The proof of eqn.(C5) is given as the following: Firstly the N µν times of braiding can be equivalently viewed as a successive braiding of N µν identical loops.As the N µν times of braiding eliminates the difference between different fusion channels, the N µν loops as a whole is actually an Abelian object, as shown in Fig. 8.And the remaining proof is similar as the Fig. 6 in Ref 1 .
Notice that the whole argument does not violate the conservation of anyon charge, as we have only specified the fusion channels but not the total charge of the initial state.And the exchanging operator for a loop with its anti-loop, i.e. the R-operator in the vacuum fusion channel, has a similar property if we do the exchanging processes for Ñµ times: ) Ñµ (C6) where although the fusion channels of the two ξ 1 µ loops or two ξ 2 µ loops are both 0, the total fusion outcome of the four loops may not be 0, i.e. generally the righthand side of (C6) should be (R σ ) ) Ñµ due to the Ñµ times of exchanging, we can write (R 0 σ ) ) Ñµ at the right-hand side of (C6) safely.

Linear Properties of the Topological Invariants
The linear properties of the braiding processes that are useful in proving the newly involved 3D constraints are: C9) which means that all the topological invariants are linear under "vertical" fusions.We firstly prove (C8) as the following: The the right-hand side of (C8) is: where we have applied the eqn.(C5), and can be any of the diagonalized entry in the matrix after fusion (C4) introduced above.The left-hand side of (C8) is: where B δ ′′ ξµξν ,(e 1 σ +e 2 σ ) can also be any of the entry in the same diagonalized matrix.Then the difference between σ ) can be eliminiated by the N µν times of braiding, which is shown in eqn.( 15) of Ref 6 .
Then (C7) and (C9) can be proved similarly, as Θ µ,σ and Θ µνλ,σ are also defined so as to eliminate the effect caused by difference charge attachments.

Partial Proof of the Constraints
We can rigorously prove the following constraints: We firstly prove (C13) as the following: By the property (C8), we have: where we realize the phase N σ Θ µν,σ by constructing N σ identical three-loop systems, and then applying the "vertical" fusions, by which the N σ type-σ base loops all together vanish.And by (C7) and (C9), (C12) and (C14) can be proved similarly.
Appendix D: Solving the Constraints

Category (B)
For simplicity, we only consider symmetry groups with order being power of 2.
b. m is even , the generating phases for the sets (B1), (B2), and (B3) are: ) By the constraint Θ 0i,0 = −Θ 0,i , a = 0, 2 (mod 4).And the remaining generating phase is determined by integer a. Hence in this case the classification is Z 2 , which belongs to BSPT phases.
(3) If 4 ≤ N i < N 0 , the generating phases for the sets (B1), (B2), and (B3) are: Ni b, we have: Combining all the constraints: The generating phases are determined by the integers: Hence in this case the classification is: where the indicators of the complex fermion layer are: (4) If 4 ≤ N 0 ≤ N i , the generating phases for the sets (B1), (B2), and (B3) are: N0 a, where when N 0 = 4 or N i ≥ N 2 0 /4, the right-hand side becomes 0 (mod 2π).Then we have: The generating phases are determined by the integers: Hence in this case the classification is where the indicators of the complex fermion layer are: For m is even, combining the cases (1)(2) into (3)(4), in conclusion the classification is: which means that the BSPT classification is Z min{Ni,N0/2} × Z min{Ni,N0/2}/2 , and a Z 2 complex fermion layer will be absorbed in the BSPT layer when N i < N 0 /2.
Hence in this case the classification is Z 2 × Z 2 , which belongs to BSPT. ( , the generating phases for the sets (D1), (D2) and (D3) are: Hence in this case the classification is Z 4 × Z 2 , which is a Z 2 non-Abelian complex fermion layer absorbed into a Z 2 × Z 2 BSPT layer. ( Combine the two constraints, the remaining three generating phases are: a (mod Hence in this case the classification is Combine the two constraints, the remaining three generating phases are: a (mod Hence in this case the classification is Appendix E: Classification of 3D FSPT phases with unitary finite Abelian G f using general group super-cohomology theory In this section, we will derive the classification of 3D FSPT with unitary finite Abelian G f , using the general group super-cohomology theory 11,18,19 .We first give a short review of general group super-cohomology theory of FSPT phases in section E 1.Some useful group cohomology calculations and relations for cocycles are given in section E 2. After that, the detailed calculations are given in sections E 3 and E 4 for non-extended and extended unitary finite Abelian G f FSPT, respectively.
All the classification data is defined modulo trivialization subgroup Γ i .For state labelled by these data, we can construct a symmetric gapped state without topological order on their boundary.Therefore, they are in fact trivial FSPT states 29 .For unitary Abelian G f , the trivialization groups Γ i can be calculated from In the rest of this section, we will use the above equations to derive a complete classification for unitary finite Abelian G f FSPT phases.

Cohomology groups, explicit cocycles and Bockstein homomorphism
There are many useful relations for cocycles of the cyclic group Z N .They can tremendously simplify the FSPT calculations.In the following, we denote the cyclic group as Z N = {0, 1, ..., N − 1}, where the group multiplication is given by the addition of integers mod N .We will use the notation = means an equality up to Z n -valued coboundaries (a.k.a.they belong to the same Z n -valued cohomology class).The symbol ⌊x⌋ is the floor function as the largest integer smaller than or equal to x.And [x] N is defined to be the mod N value of an integer x.

c. Bockstein homomorphism
The notion of Bockstein homomorphism is very useful in checking whether a cocycle (−1) f k [f k ∈ H k (G b , Z 2 )] is a U (1)-valued coboundary or not.It is defined as a mapping from H k (G b , Z 2 ) to H k+1 (G b , Z): where f k is a cocycle in H k (G b , Z 2 ).The coboundary operator is defined with appropriate plus and minus signs in integers.Because of df k 2 = 0, the right-hand-side of Eq. (E19) is always an integer.The Bockstein homomorphism is the connecting isomorphism between H k (G b , Z 2 ) and H k+1 (G b , Z).So we have the useful relation We can use it to check whether (−1) f k is a U(1)-valued coboundary or not.When acting on the cup product of two cocycles, the Bockstein homomorphism reads which is essentially the Leibniz's rule for coboundary operators.When modulo two, the Bockstein homomorphism is related to Steenrod square operation and higher cup product as There are also some useful equations for Steenrod squares: Sq i (x) The last equation is called Cartan formula.
For the special case of cohomology for group Z N , the Bockstein homomorphism of the generator n Z2 2 ∈ H 2 (Z N , Z 2 ) = Z 2 can be shown to be zero: where we used d 2 = 0 in the last step.

Classification of FSPT with
The fermionic symmetry group G f = Z f 2 × K i=1 Z Ni is associated with bosonic symmetry group and trivial central extension ω 2 = 0.It is known that, for a given positive integer N , we have a unique factorization N = p p np (p is a prime number and n p is a positive integer in Z + ) and a group isomorphism Z N ∼ = p Z p np .For prime number p > 2, the cohomology group H * (Z p np , Z 2 ) is trivial.Therefore, the symmetry group p>2 Z p np can only protect bosonic SPT phases, and can only affect the FSPT classifications though adding some BSPT phases.To understand genuine FSPT, we can assume in the bosonic symmetry group G b Eq. (E28).Without loss of generality, we can also reorder the Abelian groups such that Using Künneth formula and universal coefficient theorem, the relevant cohomology groups with Z 2 and U (1) coefficients for Eq.(E28) (N i = 2 ki ≥ 2, N i ≤ N i+1 ) are given by 1 ⟩, (E31) Z N ijkl , (E33) Here, is the binomial coefficient.We have listed the generators for the Z 2 coefficient cohomology groups, as well as the generators of some some relevant subgroups of Z coefficient cohomology groups.They are expressed as cup products of n Since ω 2 = 0, we have the trivialization groups Γ 2 = 0 and Γ 3 = 0 according to Eqs. (E5) and (E6).So all nontrivial obstruction-free n 2 and n 3 correspond to nontrivial FSPT states.

N0i gcd( 2 ,
Ni) Θ 0i,j ̸ = 0 (i ̸ = j) is proposed and proven in Ref.7.Combining the results in Ref.24 and Ref.7, we infer that Θ f ij,k = Θ iij,k = mΘ 0ij,k ̸ = 0 is the indicator for the non-Abelian statistics in the complex fermion layer.Finally Θ ijk,l ̸ = 0 is obviously the indicator for the BSPT layer by the definition of the topological invariant Θ µνλ,σ .However, if we consider a special example still belong to the non-Abelian statistics in BSPT layer.Hence we conclude that Θ ijk,l ̸ = 0 and {Θ f ij,k = 0, Θ 0ij,k ̸ = 0} are both the indicators for the non-Abelian statistics in BSPT layer.
e. the subset (C1) (or (C2), (C3)) in category C.Only the second generating phase corresponds to non-Abelian statistics and can be realized by a four-layer model construction: the first layer a is a charge-2 superconductor with chiral central charge − 1 2 (Ising type), the second layer b is a charge-2 superconductor with chiral central charge 0 (Abelian layer), the third layer c is a charge-8 superconductor with chiral central charge 0 (Abelian layer), and the fourth layer d is a charge-2 superconductor with chiral central charge 1 2 (Ising type).The 2D vortex ξ 0 is composited by a unit flux in layer a, a unit flux in layer b, four times of unit flux in layer c, and a unit flux in layer d, which therefore has quantum dimension 2. The vortex ξ i is composited by a unit flux in layer b and a unit flux in layer d, which therefore has quantum dimension √ 2. Similarly, the vortex ξ j is composited by a unit flux in layer c and a unit flux in layer d, which also has quantum dimension √ 2.

FIG. 5 .
FIG. 5.The diagram expression of the standard basis.

FIG. 8 .
FIG.8.The N µν loops as a whole can be viewed as an Abelian object.
layer and a Z 2 non-Abelian complex fermion layer.b. m is evenFor symmetry group Z f N0 × Z Ni × Z Nj × Z N k , the generating phases for the sets (D1), (D2) and (D3) are: specifying the complex fermion decorations to the intersection points of G b -domain walls.And the last ν 4 ∈ C 4 (G b , U (1))/B 3 (G b , U (1))/Γ 4 is the usual bosonic SPT classification data.These classification data satisfy the twisted cocycle equations:

n
= to mean equality up to mod n.Similarly, Z = emphasizes an equality in the ring of integers.And n,d

i+j=n
Sq i (x) ⌣ Sq j (y).(E26) ≤ i ≤ K), which are generating cocycles for the i-th Abelian group Z Ni in G b .In the following, all n (i) p are Z 2 -valued p-cocycles with the superscript Z 2 omitted.a. Trivialization

4 .Zω 2
Classification of FSPT with G f = Z f 2m × K i=1 ZN iWith the definitionN 0 = 2m, (E58) the fermionic symmetry group G f = Z f 2m × K i=1 Z Ni can be also written as G f = K i=0 Z Ni .It is associated with bosonic symmetry group G b = Z m × K i=1 Ni = Z N0/2 × (a, b) and E ijkl is protected by Z Ni , Z Nj , Z N k , Z N l .We note that in the classification of 3D FSPT phases, the classification group A is always trivial,.However, A is nontrivial for 2D FSPT phases.A ) If Ni 2 is odd and ×Z 2 ×Z Nj ×Z N k or Z f 2 ×Z Ni ×Z Nj ×Z N k), the generating phases for the sets (D1), (D2) and (D3) are: